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				<span dir="auto">Bessel function</span>
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				<div id="contentSub">&nbsp;&nbsp;(Redirected from <a href="http://en.wikipedia.org/w/index.php?title=Modified_Bessel_function&amp;redirect=no" title="Modified Bessel function">Modified Bessel function</a>)</div>
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				<div id="mw-content-text" dir="ltr" class="mw-content-ltr" lang="en"><p>In mathematics, <b>Bessel functions</b>, first defined by the mathematician <a href="http://en.wikipedia.org/wiki/Daniel_Bernoulli" title="Daniel Bernoulli">Daniel Bernoulli</a> and generalized by <a href="http://en.wikipedia.org/wiki/Friedrich_Bessel" title="Friedrich Bessel">Friedrich Bessel</a>, are <a href="http://en.wikipedia.org/wiki/Canonical_form" title="Canonical form">canonical</a> solutions <i>y</i>(<i>x</i>) of <b>Bessel's <a href="http://en.wikipedia.org/wiki/Differential_equation" title="Differential equation">differential equation</a></b>:</p>
<dl>
<dd><img class="tex" alt="x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} + (x^2 - \alpha^2)y = 0" src="Modified_Bessel_function_pliki/532a80bb3f175224ba6753e6426a3c9b.png"></dd>
</dl>
<p>for an arbitrary real or complex number α (the <b>order</b> of the Bessel function); the most common and important cases are for α an <a href="http://en.wikipedia.org/wiki/Integer" title="Integer">integer</a> or <a href="http://en.wikipedia.org/wiki/Half-integer" title="Half-integer">half-integer</a>.</p>
<p>Although α and −α produce the same differential equation, it is 
conventional to define different Bessel functions for these two orders 
(e.g., so that the Bessel functions are mostly smooth functions of α). 
Bessel functions are also known as <b>cylinder functions</b> or <b>cylindrical harmonics</b> because they are found in the solution to <a href="http://en.wikipedia.org/wiki/Laplace%27s_equation" title="Laplace's equation">Laplace's equation</a> in <a href="http://en.wikipedia.org/wiki/Cylindrical_coordinates" title="Cylindrical coordinates" class="mw-redirect">cylindrical coordinates</a>.</p>
<table id="toc" class="toc">
<tbody><tr>
<td>
<div id="toctitle">
<h2>Contents</h2>
<span class="toctoggle">&nbsp;[<a href="#" class="internal" id="togglelink">hide</a>]&nbsp;</span></div>
<ul>
<li class="toclevel-1 tocsection-1"><a href="#Applications_of_Bessel_function"><span class="tocnumber">1</span> <span class="toctext">Applications of Bessel function</span></a></li>
<li class="toclevel-1 tocsection-2"><a href="#Definitions"><span class="tocnumber">2</span> <span class="toctext">Definitions</span></a>
<ul>
<li class="toclevel-2 tocsection-3"><a href="#Bessel_functions_of_the_first_kind_:_J.CE.B1"><span class="tocnumber">2.1</span> <span class="toctext">Bessel functions of the first kind&nbsp;: <i>J</i><sub><i>α</i></sub></span></a>
<ul>
<li class="toclevel-3 tocsection-4"><a href="#Bessel.27s_integrals"><span class="tocnumber">2.1.1</span> <span class="toctext">Bessel's integrals</span></a></li>
<li class="toclevel-3 tocsection-5"><a href="#Relation_to_hypergeometric_series"><span class="tocnumber">2.1.2</span> <span class="toctext">Relation to hypergeometric series</span></a></li>
<li class="toclevel-3 tocsection-6"><a href="#Relation_to_Laguerre_polynomials"><span class="tocnumber">2.1.3</span> <span class="toctext">Relation to Laguerre polynomials</span></a></li>
</ul>
</li>
<li class="toclevel-2 tocsection-7"><a href="#Bessel_functions_of_the_second_kind_:_Y.CE.B1"><span class="tocnumber">2.2</span> <span class="toctext">Bessel functions of the second kind&nbsp;: <i>Y</i><sub><i>α</i></sub></span></a></li>
<li class="toclevel-2 tocsection-8"><a href="#Hankel_functions:_H.CE.B1.281.29.2C_H.CE.B1.282.29"><span class="tocnumber">2.3</span> <span class="toctext">Hankel functions: <i>H</i><sub><i>α</i></sub><sup>(1)</sup>, <i>H</i><sub><i>α</i></sub><sup>(2)</sup></span></a></li>
<li class="toclevel-2 tocsection-9"><a href="#Modified_Bessel_functions_:_I.CE.B1.2C_K.CE.B1"><span class="tocnumber">2.4</span> <span class="toctext">Modified Bessel functions&nbsp;: <i>I</i><sub><i>α</i></sub>, <i>K</i><sub><i>α</i></sub></span></a></li>
<li class="toclevel-2 tocsection-10"><a href="#Spherical_Bessel_functions:_jn.2C_yn"><span class="tocnumber">2.5</span> <span class="toctext">Spherical Bessel functions: <i>j</i><sub><i>n</i></sub>, <i>y</i><sub><i>n</i></sub></span></a>
<ul>
<li class="toclevel-3 tocsection-11"><a href="#Generating_function"><span class="tocnumber">2.5.1</span> <span class="toctext">Generating function</span></a></li>
<li class="toclevel-3 tocsection-12"><a href="#Differential_relations"><span class="tocnumber">2.5.2</span> <span class="toctext">Differential relations</span></a></li>
</ul>
</li>
<li class="toclevel-2 tocsection-13"><a href="#Spherical_Hankel_functions:_hn"><span class="tocnumber">2.6</span> <span class="toctext">Spherical Hankel functions: <i>h</i><sub><i>n</i></sub></span></a></li>
<li class="toclevel-2 tocsection-14"><a href="#Riccati.E2.80.93Bessel_functions:_Sn.2C_Cn.2C_.CE.BEn.2C_.CE.B6n"><span class="tocnumber">2.7</span> <span class="toctext">Riccati–Bessel functions: <i>S</i><sub><i>n</i></sub>, <i>C</i><sub><i>n</i></sub>, <i>ξ</i><sub><i>n</i></sub>, <i>ζ</i><sub><i>n</i></sub></span></a></li>
</ul>
</li>
<li class="toclevel-1 tocsection-15"><a href="#Asymptotic_forms"><span class="tocnumber">3</span> <span class="toctext">Asymptotic forms</span></a></li>
<li class="toclevel-1 tocsection-16"><a href="#Properties"><span class="tocnumber">4</span> <span class="toctext">Properties</span></a></li>
<li class="toclevel-1 tocsection-17"><a href="#Multiplication_theorem"><span class="tocnumber">5</span> <span class="toctext">Multiplication theorem</span></a></li>
<li class="toclevel-1 tocsection-18"><a href="#Bourget.27s_hypothesis"><span class="tocnumber">6</span> <span class="toctext">Bourget's hypothesis</span></a></li>
<li class="toclevel-1 tocsection-19"><a href="#Selected_identities"><span class="tocnumber">7</span> <span class="toctext">Selected identities</span></a></li>
<li class="toclevel-1 tocsection-20"><a href="#See_also"><span class="tocnumber">8</span> <span class="toctext">See also</span></a></li>
<li class="toclevel-1 tocsection-21"><a href="#Notes"><span class="tocnumber">9</span> <span class="toctext">Notes</span></a></li>
<li class="toclevel-1 tocsection-22"><a href="#References"><span class="tocnumber">10</span> <span class="toctext">References</span></a></li>
<li class="toclevel-1 tocsection-23"><a href="#External_links"><span class="tocnumber">11</span> <span class="toctext">External links</span></a></li>
</ul>
</td>
</tr>
</tbody></table>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Bessel_function&amp;action=edit&amp;section=1" title="Edit section: Applications of Bessel function">edit</a>]</span> <span class="mw-headline" id="Applications_of_Bessel_function">Applications of Bessel function</span></h2>
<p>Bessel's equation arises when finding separable solutions to <a href="http://en.wikipedia.org/wiki/Laplace%27s_equation" title="Laplace's equation">Laplace's equation</a> and the <a href="http://en.wikipedia.org/wiki/Helmholtz_equation" title="Helmholtz equation">Helmholtz equation</a> in cylindrical or <a href="http://en.wikipedia.org/wiki/Spherical_coordinates" title="Spherical coordinates" class="mw-redirect">spherical coordinates</a>. Bessel functions are therefore especially important for many problems of <a href="http://en.wikipedia.org/wiki/Wave_propagation" title="Wave propagation">wave propagation</a>
 and static potentials. In solving problems in cylindrical coordinate 
systems, one obtains Bessel functions of integer order (α = <i>n</i>); in spherical problems, one obtains half-integer orders (α = <i>n</i>&nbsp;+&nbsp;½). For example:</p>
<ul>
<li><a href="http://en.wikipedia.org/wiki/Electromagnetic_radiation" title="Electromagnetic radiation">Electromagnetic waves</a> in a cylindrical <a href="http://en.wikipedia.org/wiki/Waveguide" title="Waveguide">waveguide</a></li>
<li><a href="http://en.wikipedia.org/wiki/Conduction_%28heat%29" title="Conduction (heat)" class="mw-redirect">Heat conduction</a> in a cylindrical object</li>
<li>Modes of vibration of a thin circular (or annular) <a href="http://en.wikipedia.org/wiki/Acoustic_membrane" title="Acoustic membrane">artificial membrane</a> (such as a <a href="http://en.wikipedia.org/wiki/Drum" title="Drum">drum</a> or other <a href="http://en.wikipedia.org/wiki/Membranophone" title="Membranophone">membranophone</a>)</li>
<li>Diffusion problems on a lattice</li>
<li>Solutions to the radial <a href="http://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger equation</a> (in spherical and cylindrical coordinates) for a free particle</li>
<li>Solving for patterns of acoustical radiation</li>
</ul>
<p>Bessel functions also have useful properties for other problems, such as signal processing (e.g., see <a href="http://en.wikipedia.org/wiki/FM_synthesis" title="FM synthesis" class="mw-redirect">FM synthesis</a>, <a href="http://en.wikipedia.org/wiki/Kaiser_window" title="Kaiser window">Kaiser window</a>, or <a href="http://en.wikipedia.org/wiki/Bessel_filter" title="Bessel filter">Bessel filter</a>).</p>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Bessel_function&amp;action=edit&amp;section=2" title="Edit section: Definitions">edit</a>]</span> <span class="mw-headline" id="Definitions">Definitions</span></h2>
<p>Since this is a second-order differential equation, there must be two <a href="http://en.wikipedia.org/wiki/Linearly_independent" title="Linearly independent" class="mw-redirect">linearly independent</a>
 solutions. Depending upon the circumstances, however, various 
formulations of these solutions are convenient, and the different 
variations are described below.</p>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Bessel_function&amp;action=edit&amp;section=3" title="Edit section: Bessel functions of the first kind&nbsp;: Jα">edit</a>]</span> <span class="mw-headline" id="Bessel_functions_of_the_first_kind_:_J.CE.B1">Bessel functions of the first kind&nbsp;: <i>J</i><sub><i>α</i></sub></span></h3>
<p>Bessel functions of the first kind, denoted as <i>J</i><sub>α</sub>(<i>x</i>), are solutions of Bessel's differential equation that are finite at the origin&nbsp;(<i>x</i>&nbsp;=&nbsp;0) for integer α, and diverge as <i>x</i> approaches zero for negative non-integer&nbsp;α. The solution type (e.g., integer or non-integer) and normalization of <i>J</i><sub>α</sub>(<i>x</i>) are defined by its <a href="http://en.wikipedia.org/wiki/Bessel_function#Properties" title="Bessel function">properties</a> below. It is possible to define the function by its <a href="http://en.wikipedia.org/wiki/Taylor_series" title="Taylor series">Taylor series</a> expansion around <i>x</i>&nbsp;=&nbsp;0:<sup id="cite_ref-0" class="reference"><a href="#cite_note-0"><span>[</span>1<span>]</span></a></sup></p>
<dl>
<dd><img class="tex" alt=" J_\alpha(x) = \sum_{m=0}^\infty \frac{(-1)^m}{m! \, \Gamma(m+\alpha+1)} {\left(\tfrac{1}{2}x\right)}^{2m+\alpha} " src="Modified_Bessel_function_pliki/1b23400208b273377e8cdec7d82f0242.png"></dd>
</dl>
<p>where Γ(<i>z</i>) is the <a href="http://en.wikipedia.org/wiki/Gamma_function" title="Gamma function">gamma function</a>, a generalization of the <a href="http://en.wikipedia.org/wiki/Factorial" title="Factorial">factorial</a>
 function to non-integer values. The graphs of Bessel functions look 
roughly like oscillating sine or cosine functions that decay 
proportionally to 1/√<i>x</i> (see also their asymptotic forms below), although their roots are not generally periodic, except asymptotically for large <i>x</i>. (The Taylor series indicates that −<i>J</i><sub>1</sub>(<i>x</i>) is the derivative of <i>J</i><sub>0</sub>(<i>x</i>), much like −sin&nbsp;<i>x</i> is the derivative of cos&nbsp;<i>x</i>; more generally, the derivative of <i>J</i><sub><i>n</i></sub>(<i>x</i>) can be expressed in terms of <i>J</i><sub><i>n</i>±1</sub>(<i>x</i>) by the identities <a href="http://en.wikipedia.org/wiki/Bessel_function#Properties" title="Bessel function">below</a>.)</p>
<div class="thumb tright">
<div class="thumbinner" style="width:302px;"><a href="http://en.wikipedia.org/wiki/File:Bessel_Functions_%281st_Kind,_n%3D0,1,2%29.svg" class="image"><img alt="" src="Modified_Bessel_function_pliki/300px-Bessel_Functions_1st_Kind_n012.png" class="thumbimage" width="300" height="228"></a>
<div class="thumbcaption">
<div class="magnify"><a href="http://en.wikipedia.org/wiki/File:Bessel_Functions_%281st_Kind,_n%3D0,1,2%29.svg" class="internal" title="Enlarge"><img src="Modified_Bessel_function_pliki/magnify-clip.png" alt="" width="15" height="11"></a></div>
Plot of Bessel function of the first kind, <i>J</i><sub><i>α</i></sub>(<i>x</i>), for integer orders <i>α</i>&nbsp;=&nbsp;0,&nbsp;1,&nbsp;2.</div>
</div>
</div>
<p>For non-integer α, the functions <i>J</i><sub>α</sub>(<i>x</i>) and <i>J</i><sub>−α</sub>(<i>x</i>)
 are linearly independent, and are therefore the two solutions of the 
differential equation. On the other hand, for integer order α, the 
following relationship is valid (note that the Gamma function becomes 
infinite for negative integer arguments):<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span>[</span>2<span>]</span></a></sup></p>
<dl>
<dd><img class="tex" alt="J_{-n}(x) = (-1)^n J_{n}(x).\," src="Modified_Bessel_function_pliki/46b49cac6319f0b0ebec4e9d83303204.png"></dd>
</dl>
<p>This means that the two solutions are no longer linearly independent.
 In this case, the second linearly independent solution is then found to
 be the Bessel function of the second kind, as discussed below.</p>
<h4><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Bessel_function&amp;action=edit&amp;section=4" title="Edit section: Bessel's integrals">edit</a>]</span> <span class="mw-headline" id="Bessel.27s_integrals">Bessel's integrals</span></h4>
<p>Another definition of the Bessel function, for integer values of <img class="tex" alt="n" src="Modified_Bessel_function_pliki/7b8b965ad4bca0e41ab51de7b31363a1.png">, is possible using an integral representation:</p>
<dl>
<dd><img class="tex" alt="J_n(x) = \frac{1}{\pi} \int_0^\pi \cos (n \tau - x \sin \tau) \,\mathrm{d}\tau." src="Modified_Bessel_function_pliki/55a8c8c3474a6c254d069c72ccbd4f26.png"></dd>
</dl>
<p>Another integral representation is:</p>
<dl>
<dd><img class="tex" alt="J_n (x) = \frac{1}{2 \pi} \int_{-\pi}^\pi e^{-\mathrm{i}\,(n \tau - x \sin \tau)} \,\mathrm{d}\tau." src="Modified_Bessel_function_pliki/884d4b2760e1c4c4294e539eefcd272d.png"></dd>
</dl>
<p>This was the approach that Bessel used, and from this definition he 
derived several properties of the function. The definition may be 
extended to non-integer orders by (for <img class="tex" alt="\Re(x) &gt; 0" src="Modified_Bessel_function_pliki/09160ddf0f6d38dc07e4f536dddc0ef9.png">)</p>
<dl>
<dd><img class="tex" alt="J_\alpha(x) =
   \frac{1}{\pi} \int_0^\pi \cos(\alpha\tau- x \sin\tau)\,d\tau

 - \frac{\sin(\alpha\pi)}{\pi} \int_0^\infty
          e^{-x \sinh(t) - \alpha t} \, dt, " src="Modified_Bessel_function_pliki/6b62add89f9475ffa941df8bcf963550.png"><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span>[</span>3<span>]</span></a></sup><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span>[</span>4<span>]</span></a></sup><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span>[</span>5<span>]</span></a></sup><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span>[</span>6<span>]</span></a></sup></dd>
</dl>
<p>or for <img class="tex" alt="\alpha &gt; -\frac{1}{2}" src="Modified_Bessel_function_pliki/910cd3fa639b01457cf3967222bd9c4a.png"> by<sup class="Template-Fact" style="white-space:nowrap;">[<i><a href="http://en.wikipedia.org/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources from August 2011">citation needed</span></a></i>]</sup></p>
<dl>
<dd><img class="tex" alt="
  J_\alpha(x)= \frac{1}{2^{\alpha-1}\Gamma(\alpha + \frac{1}{2}) \sqrt{\pi}\, x^\alpha} \int_0^x (x^2-\tau^2)^{\alpha-1/2}\cos \tau \, d\tau.
" src="Modified_Bessel_function_pliki/d3b2c3e291e1f95536fd186b6fbca735.png"></dd>
</dl>
<h4><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Bessel_function&amp;action=edit&amp;section=5" title="Edit section: Relation to hypergeometric series">edit</a>]</span> <span class="mw-headline" id="Relation_to_hypergeometric_series">Relation to hypergeometric series</span></h4>
<p>The Bessel functions can be expressed in terms of the <a href="http://en.wikipedia.org/wiki/Generalized_hypergeometric_series" title="Generalized hypergeometric series" class="mw-redirect">generalized hypergeometric series</a> as<sup class="Template-Fact" style="white-space:nowrap;">[<i><a href="http://en.wikipedia.org/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources from August 2011">citation needed</span></a></i>]</sup></p>
<dl>
<dd><img class="tex" alt="J_\alpha(x)=\frac{(x/2)^\alpha}{\Gamma(\alpha+1)}  \;_0F_1 (\alpha+1; -\tfrac{1}{4}x^2)." src="Modified_Bessel_function_pliki/63c33eaaa0898cf5d0bc6a277cd5a6e6.png"></dd>
</dl>
<p>This expression is related to the development of Bessel functions in terms of the <a href="http://en.wikipedia.org/wiki/Bessel%E2%80%93Clifford_function" title="Bessel–Clifford function">Bessel–Clifford function</a>.</p>
<h4><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Bessel_function&amp;action=edit&amp;section=6" title="Edit section: Relation to Laguerre polynomials">edit</a>]</span> <span class="mw-headline" id="Relation_to_Laguerre_polynomials">Relation to Laguerre polynomials</span></h4>
<p>In terms of the <a href="http://en.wikipedia.org/wiki/Laguerre_polynomials" title="Laguerre polynomials">Laguerre polynomials</a> <img class="tex" alt="L_k" src="Modified_Bessel_function_pliki/5f81fabb8087d805e9fdd4795b135bcc.png"> and arbitrarily chosen parameter <img class="tex" alt="t" src="Modified_Bessel_function_pliki/e358efa489f58062f10dd7316b65649e.png">, the Bessel function can be expressed as<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span>[</span>7<span>]</span></a></sup></p>
<dl>
<dd><img class="tex" alt="\frac{J_\alpha(x)}{\left( \frac{x}{2}\right)^\alpha}= \frac{e^{-t}}{\Gamma(\alpha+1)} \sum_{k=0} \frac{L_k^{(\alpha)}\left( \frac{x^2}{4 t}\right)}{{k+ \alpha \choose k}} \frac{t^k}{k!}." src="Modified_Bessel_function_pliki/dc09ac91c28f1af9165947ecfc56b40b.png"></dd>
</dl>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Bessel_function&amp;action=edit&amp;section=7" title="Edit section: Bessel functions of the second kind&nbsp;: Yα">edit</a>]</span> <span class="mw-headline" id="Bessel_functions_of_the_second_kind_:_Y.CE.B1">Bessel functions of the second kind&nbsp;: <i>Y</i><sub><i>α</i></sub></span></h3>
<p>The Bessel functions of the second kind, denoted by <i>Y</i><sub><i>α</i></sub>(<i>x</i>), are solutions of the Bessel differential equation. They have a singularity at the origin (<i>x</i> = 0).</p>
<div class="thumb tright">
<div class="thumbinner" style="width:302px;"><a href="http://en.wikipedia.org/wiki/File:Bessel_Functions_%282nd_Kind,_n%3D0,1,2%29.svg" class="image"><img alt="" src="Modified_Bessel_function_pliki/300px-Bessel_Functions_2nd_Kind_n012.png" class="thumbimage" width="300" height="228"></a>
<div class="thumbcaption">
<div class="magnify"><a href="http://en.wikipedia.org/wiki/File:Bessel_Functions_%282nd_Kind,_n%3D0,1,2%29.svg" class="internal" title="Enlarge"><img src="Modified_Bessel_function_pliki/magnify-clip.png" alt="" width="15" height="11"></a></div>
Plot of Bessel function of the second kind, <i>Y</i><sub><i>α</i></sub>(<i>x</i>), for integer orders α = 0, 1, 2.</div>
</div>
</div>
<p><i>Y</i><sub><i>α</i></sub>(<i>x</i>) is sometimes also called the <b>Neumann function</b>, and is occasionally denoted instead by <i>N</i><sub><i>α</i></sub>(<i>x</i>). For non-integer α, it is related to <i>J</i><sub><i>α</i></sub>(<i>x</i>) by:</p>
<dl>
<dd><img class="tex" alt="Y_\alpha(x) = \frac{J_\alpha(x) \cos(\alpha\pi) - J_{-\alpha}(x)}{\sin(\alpha\pi)}." src="Modified_Bessel_function_pliki/02c38389b2a52132a588e3def4871872.png"></dd>
</dl>
<p>In the case of integer order <i>n</i>, the function is defined by taking the limit as a non-integer α tends to 'n':</p>
<dl>
<dd><img class="tex" alt="Y_n(x) = \lim_{\alpha \to n} Y_\alpha(x)." src="Modified_Bessel_function_pliki/e10ada75876a70970e7a67043cab2ca8.png"></dd>
</dl>
<p>There is also a corresponding integral formula (for <img class="tex" alt="\Re \{x\} &gt; 0" src="Modified_Bessel_function_pliki/9416e524e53421defb7c307dc3f6c7d7.png">),<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span>[</span>8<span>]</span></a></sup></p>
<dl>
<dd><img class="tex" alt="Y_n(x) =
   \frac{1}{\pi} \int_0^\pi \sin(x \sin\theta - n\theta) \, d\theta

 - \frac{1}{\pi} \int_0^\infty
          \left[ e^{n t} + (-1)^n e^{-n t} \right]
          e^{-x \sinh t} \, dt. " src="Modified_Bessel_function_pliki/4b4e06a3a791cec6146061e5027e3873.png"></dd>
</dl>
<p><i>Y</i><sub><i>α</i></sub>(<i>x</i>) is necessary as the second linearly independent solution of the Bessel's equation when <i>α</i> is an integer. But <i>Y</i><sub><i>α</i></sub>(<i>x</i>) has more meaning than that. It can be considered as a 'natural' partner of <i>J</i><sub><i>α</i></sub>(<i>x</i>). See also the subsection on Hankel functions below.</p>
<p>When α is an integer, moreover, as was similarly the case for the 
functions of the first kind, the following relationship is valid:</p>
<dl>
<dd><img class="tex" alt="Y_{-n}(x) = (-1)^n Y_n(x).\," src="Modified_Bessel_function_pliki/7717ab7452bf57d290f911f4b5066841.png"></dd>
</dl>
<p>Both <i>J</i><sub>α</sub>(<i>x</i>) and <i>Y</i><sub>α</sub>(<i>x</i>) are <a href="http://en.wikipedia.org/wiki/Holomorphic_function" title="Holomorphic function">holomorphic functions</a> of <i>x</i> on the <a href="http://en.wikipedia.org/wiki/Complex_plane" title="Complex plane">complex plane</a> cut along the negative real axis. When α is an integer, the Bessel functions <i>J</i> are <a href="http://en.wikipedia.org/wiki/Entire_function" title="Entire function">entire functions</a> of <i>x</i>. If <i>x</i> is held fixed, then the Bessel functions are entire functions of α.</p>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Bessel_function&amp;action=edit&amp;section=8" title="Edit section: Hankel functions: Hα(1), Hα(2)">edit</a>]</span> <span class="mw-headline" id="Hankel_functions:_H.CE.B1.281.29.2C_H.CE.B1.282.29">Hankel functions: <i>H</i><sub><i>α</i></sub><sup>(1)</sup>, <i>H</i><sub><i>α</i></sub><sup>(2)</sup></span></h3>
<p>Another important formulation of the two linearly independent solutions to Bessel's equation are the <b>Hankel functions</b> <i>H</i><sub><i>α</i></sub><sup>(1)</sup>(<i>x</i>) and <i>H</i><sub><i>α</i></sub><sup>(2)</sup>(<i>x</i>), defined by:<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span>[</span>9<span>]</span></a></sup></p>
<dl>
<dd><img class="tex" alt="H_\alpha^{(1)}(x) = J_\alpha(x) + i Y_\alpha(x)" src="Modified_Bessel_function_pliki/cd4e504d616204033371ddfbd38b06ee.png"></dd>
</dl>
<dl>
<dd><img class="tex" alt="H_\alpha^{(2)}(x) = J_\alpha(x) - i Y_\alpha(x)" src="Modified_Bessel_function_pliki/7a3b320a5139d65f84245a2f5befc70e.png"></dd>
</dl>
<p>where <i>i</i> is the <a href="http://en.wikipedia.org/wiki/Imaginary_unit" title="Imaginary unit">imaginary unit</a>. These linear combinations are also known as <b>Bessel functions of the third kind</b>; they are two linearly independent solutions of Bessel's differential equation. They are named after <a href="http://en.wikipedia.org/wiki/Hermann_Hankel" title="Hermann Hankel">Hermann Hankel</a>.</p>
<p>The importance of Hankel functions of the first and second kind lies 
more in theoretical development rather than in application. These forms 
of linear combination satisfy numerous simple-looking properties, like 
asymptotic formulae or integral representations. Here, 'simple' means an
 appearance of the factor of the form <img class="tex" alt="e^{if(x)}" src="Modified_Bessel_function_pliki/c38e9ed7ad9502f36fa55b8fe706ea77.png">. The Bessel function of the second kind then can be thought to naturally appear as the imaginary part of the Hankel functions.</p>
<p>The Hankel functions are used to express outward- and 
inward-propagating cylindrical wave solutions of the cylindrical wave 
equation, respectively (or vice versa, depending on the <a href="http://en.wikipedia.org/wiki/Sign_convention" title="Sign convention">sign convention</a> for the <a href="http://en.wikipedia.org/wiki/Frequency" title="Frequency">frequency</a>).</p>
<p>Using the previous relationships they can be expressed as:</p>
<dl>
<dd><img class="tex" alt="H_\alpha^{(1)} (x) = \frac{J_{-\alpha} (x) - e^{-\alpha \pi i} J_\alpha (x)}{i \sin (\alpha \pi)}" src="Modified_Bessel_function_pliki/61c8d0c90b0ab93bebd32be5ed23fee2.png"></dd>
</dl>
<dl>
<dd><img class="tex" alt="H_\alpha^{(2)} (x) = \frac{J_{-\alpha} (x) - e^{\alpha \pi i} J_\alpha (x)}{- i \sin (\alpha \pi)}" src="Modified_Bessel_function_pliki/a652ab4c4a868540d32cc5dc4f04b2fb.png"></dd>
</dl>
<p>if <i>α</i> is an integer, the limit has to be calculated. The following relationships are valid, whether <i>α</i> is an integer or not:<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span>[</span>10<span>]</span></a></sup></p>
<dl>
<dd><img class="tex" alt="H_{-\alpha}^{(1)} (x)= e^{\alpha \pi i} H_\alpha^{(1)} (x) " src="Modified_Bessel_function_pliki/29cfa67b6606ce11b91abfacf80e0d47.png"></dd>
</dl>
<dl>
<dd><img class="tex" alt="H_{-\alpha}^{(2)} (x)= e^{-\alpha \pi i} H_\alpha^{(2)} (x). " src="Modified_Bessel_function_pliki/d555041a2f809add7138d095b91d3ecc.png"></dd>
</dl>
<p>The Hankel functions admit the following integral representations for <img class="tex" alt="\Re x &gt; 0" src="Modified_Bessel_function_pliki/f5b752a5b51b98aec849ac19a913861a.png">:<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span>[</span>11<span>]</span></a></sup></p>
<dl>
<dd><img class="tex" alt="H_\alpha^{(1)} (x)= \frac{1}{\pi i}\int_{-\infty}^{+\infty+i\pi} e^{x\sinh t - \alpha t} \, dt, " src="Modified_Bessel_function_pliki/519af1e9edd7d832b8621ae3c7fa7fb2.png"></dd>
<dd><img class="tex" alt="H_\alpha^{(2)} (x)= -\frac{1}{\pi i}\int_{-\infty}^{+\infty-i\pi} e^{x\sinh t - \alpha t} \, dt, " src="Modified_Bessel_function_pliki/74e46aa2b743f3f4b2d5db06c9fbf052.png"></dd>
</dl>
<p>where the integration limits indicate integration along a <a href="http://en.wikipedia.org/wiki/Methods_of_contour_integration" title="Methods of contour integration">contour</a>
 that can be chosen as follows: from −∞ to 0 along the negative real 
axis, from 0 to ±iπ along the imaginary axis, and from ±iπ to +∞±iπ 
along a contour parallel to the real axis.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span>[</span>12<span>]</span></a></sup></p>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Bessel_function&amp;action=edit&amp;section=9" title="Edit section: Modified Bessel functions&nbsp;: Iα, Kα">edit</a>]</span> <span class="mw-headline" id="Modified_Bessel_functions_:_I.CE.B1.2C_K.CE.B1">Modified Bessel functions&nbsp;: <i>I</i><sub><i>α</i></sub>, <i>K</i><sub><i>α</i></sub></span></h3>
<p>The Bessel functions are valid even for <a href="http://en.wikipedia.org/wiki/Complex_number" title="Complex number">complex</a> arguments <i>x</i>,
 and an important special case is that of a purely imaginary argument. 
In this case, the solutions to the Bessel equation are called the <b>modified Bessel functions</b> (or occasionally the <b>hyperbolic Bessel functions</b>) <b>of the first and second kind</b>, and are defined by any of these equivalent alternatives:<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span>[</span>13<span>]</span></a></sup></p>
<dl>
<dd><img class="tex" alt="I_\alpha(x) = i^{-\alpha} J_\alpha(ix) =\sum_{m=0}^\infty \frac{1}{m! \Gamma(m+\alpha+1)}\left(\frac{x}{2}\right)^{2m+\alpha}" src="Modified_Bessel_function_pliki/efd9e6e7789dc2172798b6fda44cc8f1.png"></dd>
</dl>
<dl>
<dd><img class="tex" alt="K_\alpha(x) = \frac{\pi}{2} \frac{I_{-\alpha} (x) - I_\alpha (x)}{\sin (\alpha \pi)} = \frac{\pi}{2} i^{\alpha+1} H_\alpha^{(1)}(ix) = \frac{\pi}{2} (-i)^{\alpha+1} H_\alpha^{(2)}(-ix)." src="Modified_Bessel_function_pliki/c36cf06cd53aae81c0c0ec06d2e236e8.png"></dd>
</dl>
<p>These are chosen to be real-valued for real and positive arguments <i>x</i>. The series expansion for <i>I<sub>α</sub></i>(<i>x</i>) is thus similar to that for <i>J<sub>α</sub></i>(<i>x</i>), but without the alternating (−1)<sup><i>m</i></sup> factor.</p>
<p><i>I<sub>α</sub></i>(<i>x</i>) and <i>K<sub>α</sub></i>(<i>x</i>) are the two linearly independent solutions to the modified Bessel's equation:<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span>[</span>14<span>]</span></a></sup></p>
<dl>
<dd><img class="tex" alt="x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} - (x^2 + \alpha^2)y = 0." src="Modified_Bessel_function_pliki/3812040e56e8949d718964443e3fe282.png"></dd>
</dl>
<p>Unlike the ordinary Bessel functions, which are oscillating as functions of a real argument, <i>I</i><sub>α</sub> and <i>K</i><sub>α</sub> are <a href="http://en.wikipedia.org/wiki/Exponential_growth" title="Exponential growth">exponentially growing</a> and <a href="http://en.wikipedia.org/wiki/Exponential_decay" title="Exponential decay">decaying</a> functions, respectively. Like the ordinary Bessel function <i>J</i><sub>α</sub>, the function <i>I</i><sub>α</sub> goes to zero at <i>x</i> = 0 for α &gt; 0 and is finite at <i>x</i> = 0 for α = 0. Analogously, <i>K</i><sub>α</sub> diverges at <i>x</i>&nbsp;=&nbsp;0.</p>
<table align="center">
<tbody><tr>
<td>
<div class="thumb tnone">
<div class="thumbinner" style="width:302px;"><a href="http://en.wikipedia.org/wiki/File:BesselI_Functions_%281st_Kind,_n%3D0,1,2,3%29.svg" class="image"><img alt="" src="Modified_Bessel_function_pliki/300px-BesselI_Functions_1st_Kind_n0123.png" class="thumbimage" width="300" height="228"></a>
<div class="thumbcaption">
<div class="magnify"><a href="http://en.wikipedia.org/wiki/File:BesselI_Functions_%281st_Kind,_n%3D0,1,2,3%29.svg" class="internal" title="Enlarge"><img src="Modified_Bessel_function_pliki/magnify-clip.png" alt="" width="15" height="11"></a></div>
Modified Bessel functions of 1st kind, <i>I</i><sub><i>α</i></sub>(<i>x</i>), for <i>α</i>&nbsp;=&nbsp;0,&nbsp;1,&nbsp;2,&nbsp;3</div>
</div>
</div>
</td>
<td>
<div class="thumb tnone">
<div class="thumbinner" style="width:302px;"><a href="http://en.wikipedia.org/wiki/File:BesselK_Functions_%28n%3D0,1,2,3%29.svg" class="image"><img alt="" src="Modified_Bessel_function_pliki/300px-BesselK_Functions_n0123.png" class="thumbimage" width="300" height="228"></a>
<div class="thumbcaption">
<div class="magnify"><a href="http://en.wikipedia.org/wiki/File:BesselK_Functions_%28n%3D0,1,2,3%29.svg" class="internal" title="Enlarge"><img src="Modified_Bessel_function_pliki/magnify-clip.png" alt="" width="15" height="11"></a></div>
Modified Bessel functions of 2nd kind, <i>K</i><sub><i>α</i></sub>(<i>x</i>), for <i>α</i>&nbsp;=&nbsp;0,&nbsp;1,&nbsp;2,&nbsp;3</div>
</div>
</div>
</td>
</tr>
</tbody></table>
<p><br>
Two integral formulas for the modified Bessel functions are (for <img class="tex" alt="\Re \{x\} &gt; 0" src="Modified_Bessel_function_pliki/9416e524e53421defb7c307dc3f6c7d7.png">):<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span>[</span>15<span>]</span></a></sup></p>
<dl>
<dd><img class="tex" alt="I_\alpha(x) = \frac{1}{\pi}\int_0^\pi \exp(x\cos\theta) \cos(\alpha\theta) d\theta - \frac{\sin(\alpha\pi)}{\pi}\int_0^\infty \exp(-x\cosh t - \alpha t) dt ," src="Modified_Bessel_function_pliki/851eb4d3c84a23e63290ecca04d75403.png"></dd>
<dd><img class="tex" alt="K_\alpha(x) = \int_0^\infty \exp(-x\cosh t) \cosh(\alpha t) dt." src="Modified_Bessel_function_pliki/1b5ca07cac01d388b5c339cb69ecfe48.png"></dd>
</dl>
<p>Modified Bessel functions <img class="tex" alt=" K_{1/3}" src="Modified_Bessel_function_pliki/79dc51b02537d4cb0408c6cab4e90093.png"> and <img class="tex" alt=" K_{2/3}" src="Modified_Bessel_function_pliki/70464bc187c64c76e388f051cbc35aeb.png"> can be represented in terms of rapidly converged integrals<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span>[</span>16<span>]</span></a></sup></p>
<dl>
<dd><img class="tex" alt=" K_{1/3} (\xi) = \sqrt{3}\, \int_0^\infty \, \exp \left[- \xi
\left(1+\frac{4x^2}{3}\right) \sqrt{1+\frac{x^2}{3}} \right] \ dx " src="Modified_Bessel_function_pliki/133db3ffce7bd8555037a884ccef716d.png"></dd>
</dl>
<dl>
<dd><img class="tex" alt=" K_{2/3} (\xi) = \frac{1}{ \sqrt{3}} \,
\int_0^\infty \, \frac{3+2x^2}{\sqrt{1+x^2/3}}
\exp  \left[- \xi  \left(1+\frac{4x^2}{3}\right) \sqrt{1+\frac{x^2}{3}} \right] \ dx " src="Modified_Bessel_function_pliki/616dd664a82dd5ed48bb4ed456d3fb86.png"></dd>
</dl>
<p>The <b>modified Bessel function of the second kind</b> has also been called by the now-rare names:</p>
<ul>
<li><b>Basset function</b> after <a href="http://en.wikipedia.org/wiki/Alfred_Barnard_Basset" title="Alfred Barnard Basset">Alfred Barnard Basset</a></li>
<li><b>Modified Bessel function of the third kind</b></li>
<li><b>Modified Hankel function</b><sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span>[</span>17<span>]</span></a></sup></li>
<li><b>Macdonald function</b></li>
</ul>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Bessel_function&amp;action=edit&amp;section=10" title="Edit section: Spherical Bessel functions: jn, yn">edit</a>]</span> <span class="mw-headline" id="Spherical_Bessel_functions:_jn.2C_yn">Spherical Bessel functions: <i>j</i><sub><i>n</i></sub>, <i>y</i><sub><i>n</i></sub></span></h3>
<div class="thumb tright">
<div class="thumbinner" style="width:302px;"><a href="http://en.wikipedia.org/wiki/File:Spherical_Bessel_j_Functions_%28n%3D0,1,2%29.svg" class="image"><img alt="" src="Modified_Bessel_function_pliki/300px-Spherical_Bessel_j_Functions_n012.png" class="thumbimage" width="300" height="228"></a>
<div class="thumbcaption">
<div class="magnify"><a href="http://en.wikipedia.org/wiki/File:Spherical_Bessel_j_Functions_%28n%3D0,1,2%29.svg" class="internal" title="Enlarge"><img src="Modified_Bessel_function_pliki/magnify-clip.png" alt="" width="15" height="11"></a></div>
Spherical Bessel functions of 1st kind, <i>j</i><sub><i>n</i></sub>(<i>x</i>), for <i>n</i>&nbsp;=&nbsp;0,&nbsp;1,&nbsp;2</div>
</div>
</div>
<div class="thumb tright">
<div class="thumbinner" style="width:302px;"><a href="http://en.wikipedia.org/wiki/File:Spherical_Bessel_y_Functions_%28n%3D0,1,2%29.svg" class="image"><img alt="" src="Modified_Bessel_function_pliki/300px-Spherical_Bessel_y_Functions_n012.png" class="thumbimage" width="300" height="228"></a>
<div class="thumbcaption">
<div class="magnify"><a href="http://en.wikipedia.org/wiki/File:Spherical_Bessel_y_Functions_%28n%3D0,1,2%29.svg" class="internal" title="Enlarge"><img src="Modified_Bessel_function_pliki/magnify-clip.png" alt="" width="15" height="11"></a></div>
Spherical Bessel functions of 2nd kind, <i>y</i><sub><i>n</i></sub>(<i>x</i>), for <i>n</i> =&nbsp;0,&nbsp;1,&nbsp;2</div>
</div>
</div>
<p>When solving the <a href="http://en.wikipedia.org/wiki/Helmholtz_equation" title="Helmholtz equation">Helmholtz equation</a> in spherical coordinates by separation of variables, the radial equation has the form:</p>
<dl>
<dd><img class="tex" alt="x^2 \frac{d^2 y}{dx^2} + 2x \frac{dy}{dx} + [x^2 - n(n+1)]y = 0." src="Modified_Bessel_function_pliki/316a1e7b77811ba96493039cc95234cd.png"></dd>
</dl>
<p>The two linearly independent solutions to this equation are called the <b>spherical Bessel functions</b> <i>j</i><sub><i>n</i></sub> and <i>y</i><sub><i>n</i></sub>, and are related to the ordinary Bessel functions <i>J</i><sub><i>n</i></sub> and <i>Y</i><sub><i>n</i></sub> by:<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span>[</span>18<span>]</span></a></sup></p>
<dl>
<dd><img class="tex" alt="j_{n}(x) = \sqrt{\frac{\pi}{2x}} J_{n+1/2}(x)," src="Modified_Bessel_function_pliki/3d928f4c28983196c21378368ecc4500.png"></dd>
</dl>
<dl>
<dd><img class="tex" alt="y_{n}(x) = \sqrt{\frac{\pi}{2x}} Y_{n+1/2}(x) = (-1)^{n+1} \sqrt{\frac{\pi}{2x}} J_{-n-1/2}(x)." src="Modified_Bessel_function_pliki/eb0b089e6cf0e427d7bd019597a6725f.png"></dd>
</dl>
<p><img class="tex" alt="y_n" src="Modified_Bessel_function_pliki/0aac89cc5848912240b16f540cc5a674.png"> is also denoted <img class="tex" alt="n_n" src="Modified_Bessel_function_pliki/6efbd28cb94e3173ab6360e23b30040e.png"> or <a href="http://en.wikipedia.org/wiki/Eta_%28letter%29" title="Eta (letter)" class="mw-redirect">η</a><sub>n</sub>; some authors call these functions the <b>spherical Neumann functions</b>.</p>
<p>The spherical Bessel functions can also be written as (<b>Rayleigh's Formulas</b>):<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span>[</span>19<span>]</span></a></sup></p>
<dl>
<dd><img class="tex" alt="j_n(x) = (-x)^n \left(\frac{1}{x}\frac{d}{dx}\right)^n\,\frac{\sin x}{x} ," src="Modified_Bessel_function_pliki/cc3d1b5798724e1a9d80ec28f7e88e8d.png"></dd>
<dd><img class="tex" alt="y_n(x) = -(-x)^n \left(\frac{1}{x}\frac{d}{dx}\right)^n\,\frac{\cos x}{x}." src="Modified_Bessel_function_pliki/863d8e368555c1b922e1ebf508607bfe.png"></dd>
</dl>
<p>The first spherical Bessel function <img class="tex" alt="j_0(x)" src="Modified_Bessel_function_pliki/bf5001ec990322d2e78e07b8cf785259.png"> is also known as the (unnormalized) <a href="http://en.wikipedia.org/wiki/Sinc_function" title="Sinc function">sinc function</a>. The first few spherical Bessel functions are:</p>
<dl>
<dd><img class="tex" alt="j_0(x)=\frac{\sin x} {x}" src="Modified_Bessel_function_pliki/727d623190b90efc516a6cce2bb1bf2f.png"></dd>
<dd><img class="tex" alt="j_1(x)=\frac{\sin x} {x^2}- \frac{\cos x} {x}" src="Modified_Bessel_function_pliki/52f0e9f2f1a531270a9ba8b1ae849f8f.png"></dd>
<dd><img class="tex" alt="j_2(x)=\left(\frac{3} {x^2} - 1 \right)\frac{\sin x}{x} - \frac{3\cos x} {x^2}" src="Modified_Bessel_function_pliki/a72ceb3b9a18e464c1e206adfede4534.png"><sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span>[</span>20<span>]</span></a></sup></dd>
<dd><img class="tex" alt="j_3(x)=\left(\frac{15}{x^3} - \frac{6}{x} \right)\frac{\sin x}{x} -\left(\frac{15}{x^2} - 1\right) \frac{\cos x} {x}," src="Modified_Bessel_function_pliki/a46ba60d8a41b83e0421eb0354d0aa37.png"></dd>
</dl>
<p>and</p>
<dl>
<dd><img class="tex" alt="y_0(x)=-j_{-1}(x)=-\,\frac{\cos x} {x}" src="Modified_Bessel_function_pliki/4bf47373cc233be85246b42c9bd5125f.png"></dd>
<dd><img class="tex" alt="y_1(x)=j_{-2}(x)=-\,\frac{\cos x} {x^2}- \frac{\sin x} {x}" src="Modified_Bessel_function_pliki/e54e0a1763689716ba0398fab323d557.png"></dd>
<dd><img class="tex" alt="y_2(x)=-j_{-3}(x)=\left(-\,\frac{3}{x^2}+1 \right)\frac{\cos x}{x}- \frac{3 \sin x} {x^2}" src="Modified_Bessel_function_pliki/51764f53eecfd042fe0e31c8c7816800.png"><sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span>[</span>21<span>]</span></a></sup></dd>
<dd><img class="tex" alt="y_{3}\left( x\right)=j_{-4}(x) =\left( -\frac{15}{x^{3}}+\frac{6}{x}\right) \frac{\cos
x}{x}-\left( \frac{15}{x^{2}}-1\right) \frac{\sin x}{x}." src="Modified_Bessel_function_pliki/7f31871fd1cd509e856cba4a32283bf6.png"></dd>
</dl>
<p>The general identities are<sup class="Template-Fact" style="white-space:nowrap;">[<i><a href="http://en.wikipedia.org/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources from August 2011">citation needed</span></a></i>]</sup></p>
<dl>
<dd><img class="tex" alt="
\begin{align}
J_{n+\frac 1 2}(x)=\sqrt{\frac 2 {\pi x}}\sum_{i=0}^\frac {n+1} 2 (-1)^{n-i} &amp; \left[ \sin(x) \left(\frac 2 x\right)^{n-2i} \frac {(n-i)!}{i!} {-\frac 1 2 -i \choose n-2i} \right. \\
&amp; \left.{} - \cos(x) \left(\frac 2 x\right)^{n+1-2i} \frac {(n-i)!}{i!} i {-\frac 1 2 -i \choose n-2i+1}\right],
\end{align}
" src="Modified_Bessel_function_pliki/f17cb7f6f8b0c9bbd4de5abab1f7c81a.png"></dd>
</dl>
<p>(the upper limit of summation is understood to be the largest integer less than or equal to (n+1)/2); a closed form employing <a href="http://en.wikipedia.org/wiki/Laguerre_polynomial" title="Laguerre polynomial" class="mw-redirect">Laguerre's polynomial</a> <img class="tex" alt="L_n" src="Modified_Bessel_function_pliki/ecca3c5a6b5b7cbd578ccd803de7cfe8.png"> (cf. <a href="http://en.wikipedia.org/wiki/Bessel_polynomial" title="Bessel polynomial" class="mw-redirect">Bessel's polynomial</a>) and other representations are provided by<sup class="Template-Fact" style="white-space:nowrap;">[<i><a href="http://en.wikipedia.org/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources from August 2011">citation needed</span></a></i>]</sup></p>
<dl>
<dd><img class="tex" alt="\begin{align}I_{n+\frac 1 2}(x)&amp;=\frac{n!}{\sqrt\pi (2x)^{n+\frac 1 2}}\left(e^x\cdot L_n^{(-2n-1)}(-2x)-e^{-x}\cdot L_n^{(-2n-1)}(2x)\right)\\
&amp;=\frac{n!e^{-x}}{\sqrt\pi (2x)^{n+\frac12}}\sum_{k=2n+1}^\infty{k-n-1\choose n}\frac{(2x)^k}{k!}\\
&amp;=\frac{e^x}{\sqrt{2\pi x}}(-1)^{n+1}\sum_{k=n}^\infty \frac{(-2x)^{k+1}k!}{(k-n)!(k+n+1)!}.\end{align}" src="Modified_Bessel_function_pliki/53d48aaa970b280f1db417925dd86ba3.png"></dd>
</dl>
<h4><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Bessel_function&amp;action=edit&amp;section=11" title="Edit section: Generating function">edit</a>]</span> <span class="mw-headline" id="Generating_function">Generating function</span></h4>
<p>The spherical Bessel functions have the generating functions <sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span>[</span>22<span>]</span></a></sup></p>
<dl>
<dd><img class="tex" alt="\frac 1 {z} \cos \sqrt{z^2 - 2zt}= \sum_{n=0}^\infty \frac{t^n}{n!} j_{n-1}(z), " src="Modified_Bessel_function_pliki/85d8435ad9fca870ed6f348313a29a86.png"></dd>
<dd><img class="tex" alt="\frac 1 {z} \sin \sqrt{z^2 + 2zt}= \sum_{n=0}^\infty \frac{(-t)^n}{n!} y_{n-1}(z) ." src="Modified_Bessel_function_pliki/62018709e6751d0e1f3d9dd7d7c8588c.png"></dd>
</dl>
<h4><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Bessel_function&amp;action=edit&amp;section=12" title="Edit section: Differential relations">edit</a>]</span> <span class="mw-headline" id="Differential_relations">Differential relations</span></h4>
<p>In the following <img class="tex" alt="f_n" src="Modified_Bessel_function_pliki/566b85f27e5486bb63c0d15775fbd769.png"> is any of <img class="tex" alt="j_n, y_n, h_n^{(1)}, h_n^{(2)}" src="Modified_Bessel_function_pliki/c9e5e3f7d4fb2e95337331f5379c80c8.png"> for <img class="tex" alt="n=0,\pm 1,\pm 2,\dots" src="Modified_Bessel_function_pliki/016982435417ce9f28acb44870b155c1.png"></p>
<dl>
<dd><img class="tex" alt="\left(\frac{1}{z}\frac{d}{dz}\right)^m\left(z^{n+1}f_n(z)\right)=z^{(n-m)+1}f_{(n-m)}(z)." src="Modified_Bessel_function_pliki/769eba88e9034a2106c3930efe81083e.png"><sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span>[</span>23<span>]</span></a></sup></dd>
</dl>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Bessel_function&amp;action=edit&amp;section=13" title="Edit section: Spherical Hankel functions: hn">edit</a>]</span> <span class="mw-headline" id="Spherical_Hankel_functions:_hn">Spherical Hankel functions: <i>h</i><sub><i>n</i></sub></span></h3>
<p>There are also spherical analogues of the Hankel functions:</p>
<dl>
<dd><img class="tex" alt="h_n^{(1)}(x) = j_n(x) + i y_n(x) \, " src="Modified_Bessel_function_pliki/c1e30449c1cb2bd124158a5eb2cdafc6.png"></dd>
</dl>
<dl>
<dd><img class="tex" alt="h_n^{(2)}(x) = j_n(x) - i y_n(x). \, " src="Modified_Bessel_function_pliki/464d9756de1cae32f50dcfda449d5c4f.png"></dd>
</dl>
<p>In fact, there are simple closed-form expressions for the Bessel functions of <a href="http://en.wikipedia.org/wiki/Half-integer" title="Half-integer">half-integer</a> order in terms of the standard <a href="http://en.wikipedia.org/wiki/Trigonometric_function" title="Trigonometric function" class="mw-redirect">trigonometric functions</a>, and therefore for the spherical Bessel functions. In particular, for non-negative integers <i>n</i>:</p>
<dl>
<dd><img class="tex" alt="h_n^{(1)}(x) = (-i)^{n+1} \frac{e^{ix}}{x} \sum_{m=0}^n \frac{i^m}{m!(2x)^m} \frac{(n+m)!}{(n-m)!}" src="Modified_Bessel_function_pliki/a87981a111f5d9779b4f83bf2091ca76.png"></dd>
</dl>
<p>and <img class="tex" alt="h_n^{(2)}" src="Modified_Bessel_function_pliki/d66c593040f4c2c5ab54586115ef7cb9.png"> is the complex-conjugate of this (for real <img class="tex" alt="x" src="Modified_Bessel_function_pliki/9dd4e461268c8034f5c8564e155c67a6.png">). It follows, for example, that <img class="tex" alt="j_0(x) = \sin(x)/x" src="Modified_Bessel_function_pliki/b6b4fe714b23cb2efd51feb71641b7a9.png"> and <img class="tex" alt="y_0(x) = -\cos(x)/x" src="Modified_Bessel_function_pliki/53ebb6082945584ea9611666d4b98b63.png">, and so on.</p>
<p>The spherical Hankel functions appear in problems involving spherical wave propagation, for example in <a href="http://en.wikipedia.org/wiki/Electromagnetic_wave_equation#Multipole_expansion" title="Electromagnetic wave equation">the multipole expansion of the electromagnetic field</a>.</p>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Bessel_function&amp;action=edit&amp;section=14" title="Edit section: Riccati–Bessel functions: Sn, Cn, ξn, ζn">edit</a>]</span> <span class="mw-headline" id="Riccati.E2.80.93Bessel_functions:_Sn.2C_Cn.2C_.CE.BEn.2C_.CE.B6n">Riccati–Bessel functions: <i>S</i><sub><i>n</i></sub>, <i>C</i><sub><i>n</i></sub>, <i>ξ</i><sub><i>n</i></sub>, <i>ζ</i><sub><i>n</i></sub></span></h3>
<p><a href="http://en.wikipedia.org/wiki/Jacopo_Riccati" title="Jacopo Riccati">Riccati</a>–Bessel functions only slightly differ from spherical Bessel functions:</p>
<dl>
<dd><img class="tex" alt="S_n(x)=x j_n(x)=\sqrt{\pi x/2} \, J_{n+1/2}(x)" src="Modified_Bessel_function_pliki/0d43cbb6c17c49572cf28bc094be9f5f.png"></dd>
</dl>
<dl>
<dd><img class="tex" alt="C_n(x)=-x y_n(x)=-\sqrt{\pi x/2} \, Y_{n+1/2}(x)" src="Modified_Bessel_function_pliki/76340667e991095c50263573f0bfd5ed.png"></dd>
</dl>
<dl>
<dd><img class="tex" alt="\xi_n(x) = x h_n^{(1)}(x)=\sqrt{\pi x/2} \, H_{n+1/2}^{(1)}(x)=S_n(x)-iC_n(x)" src="Modified_Bessel_function_pliki/789c38b080d1703064cb4a15472d9b83.png"></dd>
</dl>
<dl>
<dd><img class="tex" alt="\zeta_n(x)=x h_n^{(2)}(x)=\sqrt{\pi x/2} \, H_{n+1/2}^{(2)}(x)=S_n(x)+iC_n(x)." src="Modified_Bessel_function_pliki/031e0d2b45d8b6838743e7bef4abb17f.png"></dd>
</dl>
<p>They satisfy the differential equation:</p>
<dl>
<dd><img class="tex" alt="x^2 \frac{d^2 y}{dx^2} + [x^2 - n (n+1)] y = 0." src="Modified_Bessel_function_pliki/0dc9509ffe5ae3661f508fe1752ba995.png"></dd>
</dl>
<p>This differential equation, and the Riccati–Bessel solutions, arises 
in the problem of scattering of electromagnetic waves by a sphere, known
 as <a href="http://en.wikipedia.org/wiki/Mie_scattering" title="Mie scattering">Mie scattering</a> after the first published solution by Mie (1908). See e.g., Du (2004)<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span>[</span>24<span>]</span></a></sup> for recent developments and references.</p>
<p>Following <a href="http://en.wikipedia.org/wiki/Peter_Debye" title="Peter Debye">Debye</a> (1909), the notation <img class="tex" alt="\psi_n,\chi_n" src="Modified_Bessel_function_pliki/42ceb6e99a174986c2ef75d67060d227.png"> is sometimes used instead of <img class="tex" alt="S_n,C_n" src="Modified_Bessel_function_pliki/794c40ac0c5b008ddeea95914e7d8028.png">.</p>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Bessel_function&amp;action=edit&amp;section=15" title="Edit section: Asymptotic forms">edit</a>]</span> <span class="mw-headline" id="Asymptotic_forms">Asymptotic forms</span></h2>
<p>The Bessel functions have the following <a href="http://en.wikipedia.org/wiki/Asymptotic_analysis" title="Asymptotic analysis">asymptotic</a> forms for non-negative α. For small arguments <img class="tex" alt="0 &lt; x \ll \sqrt{\alpha + 1}" src="Modified_Bessel_function_pliki/9ce7b1f99c7533f54d28e0e1ac8ad8aa.png">, one obtains:<sup id="cite_ref-Arfken_.26_Weber_24-0" class="reference"><a href="#cite_note-Arfken_.26_Weber-24"><span>[</span>25<span>]</span></a></sup></p>
<dl>
<dd><img class="tex" alt="J_\alpha(x) \approx \frac{1}{\Gamma(\alpha+1)} \left( \frac{x}{2} \right) ^\alpha " src="Modified_Bessel_function_pliki/a1c4466a8385a5704b1451cbd80aefe7.png"></dd>
</dl>
<dl>
<dd><img class="tex" alt="Y_\alpha(x) \approx \begin{cases}
  \frac{2}{\pi} \left[ \ln (x/2) + \gamma \right]  &amp; \text{if } \alpha=0 \\ \\
  -\frac{\Gamma(\alpha)}{\pi} \left( \frac{2}{x} \right) ^\alpha &amp; \text{if } \alpha &gt; 0
\end{cases} " src="Modified_Bessel_function_pliki/14916551b464b20bf8e3a19b60890c32.png"></dd>
</dl>
<p>where <img class="tex" alt="\gamma" src="Modified_Bessel_function_pliki/334de1ea38b615839e4ee6b65ee1b103.png"> is the <a href="http://en.wikipedia.org/wiki/Euler%E2%80%93Mascheroni_constant" title="Euler–Mascheroni constant">Euler–Mascheroni constant</a> (0.5772...) and <img class="tex" alt="\Gamma" src="Modified_Bessel_function_pliki/162d4c413f99ae2763b1ced17ed1a14b.png"> denotes the <a href="http://en.wikipedia.org/wiki/Gamma_function" title="Gamma function">gamma function</a>. For large arguments <img class="tex" alt="x \gg |\alpha^2 - 1/4|" src="Modified_Bessel_function_pliki/9ce96a9bd32796313ef36d122308d3ee.png">, they become:<sup id="cite_ref-Arfken_.26_Weber_24-1" class="reference"><a href="#cite_note-Arfken_.26_Weber-24"><span>[</span>25<span>]</span></a></sup></p>
<dl>
<dd><img class="tex" alt="J_\alpha(x)\approx \sqrt{\frac{2}{\pi x}}
        \cos \left( x-\frac{\alpha\pi}{2} - \frac{\pi}{4} \right)" src="Modified_Bessel_function_pliki/592e657996055ae203d19f7144fef972.png"></dd>
</dl>
<dl>
<dd><img class="tex" alt="Y_\alpha(x) \approx \sqrt{\frac{2}{\pi x}}
        \sin \left( x-\frac{\alpha\pi}{2} - \frac{\pi}{4} \right)." src="Modified_Bessel_function_pliki/931db485ad6a951aeb754722498cd91e.png"></dd>
</dl>
<p>(For α=1/2 these formulas are exact; see the spherical Bessel 
functions above.) Asymptotic forms for the other types of Bessel 
function follow straightforwardly from the above relations. For example,
 for large <img class="tex" alt="x \gg |\alpha^2 - 1/4|" src="Modified_Bessel_function_pliki/9ce96a9bd32796313ef36d122308d3ee.png">, the modified Bessel functions become:</p>
<dl>
<dd><img class="tex" alt="I_\alpha(x) \approx \frac{e^x}{\sqrt{2\pi x}} \left(1 - \frac{4 \alpha^{2} - 1}{8 x} + \frac{(4 \alpha^{2} - 1) (4 \alpha^{2} - 9)}{2! (8 x)^{2}} - \frac{(4 \alpha^{2} - 1) (4 \alpha^{2} - 9) (4 \alpha^{2} - 25)}{3! (8 x)^{3}} + \cdots \right) ," src="Modified_Bessel_function_pliki/5ff35926e566781403d3b67cfdb4d686.png"><sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span>[</span>26<span>]</span></a></sup></dd>
</dl>
<dl>
<dd><img class="tex" alt="K_\alpha(x) \approx \sqrt{\frac{\pi}{2x}} e^{-x} \left(1 + \frac{4 \alpha^{2} - 1}{8 x} + \frac{(4 \alpha^{2} - 1) (4 \alpha^{2} - 9)}{2! (8 x)^{2}} + \frac{(4 \alpha^{2} - 1) (4 \alpha^{2} - 9) (4 \alpha^{2} - 25)}{3! (8 x)^{3}} + \cdots \right) ." src="Modified_Bessel_function_pliki/85a86ef1dc4889c2571771bdeed6d2f4.png"><sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span>[</span>27<span>]</span></a></sup></dd>
</dl>
<p>Similarly, the last expressions are exact when <img class="tex" alt="\alpha=1/2" src="Modified_Bessel_function_pliki/6d9ca46bdd6fa38c2ed5b145483623d0.png">.</p>
<p>For small arguments <img class="tex" alt="0 &lt; x \ll \sqrt{\alpha + 1}" src="Modified_Bessel_function_pliki/9ce7b1f99c7533f54d28e0e1ac8ad8aa.png">, they become:</p>
<dl>
<dd><img class="tex" alt="I_\alpha(x) \approx \frac{1}{\Gamma(\alpha+1)} \left( \frac{x}{2} \right) ^\alpha " src="Modified_Bessel_function_pliki/9ea8d8858e11289ec7ace191a759c328.png"></dd>
</dl>
<dl>
<dd><img class="tex" alt="K_\alpha(x) \approx \begin{cases}
  - \ln (x/2) - \gamma   &amp; \text{if } \alpha=0 \\ \\
  \frac{\Gamma(\alpha)}{2} \left( \frac{2}{x} \right) ^\alpha &amp; \text{if } \alpha &gt; 0.
\end{cases} " src="Modified_Bessel_function_pliki/97e4daa9ba5a16ef7d9126aaa20bbf32.png"></dd>
</dl>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Bessel_function&amp;action=edit&amp;section=16" title="Edit section: Properties">edit</a>]</span> <span class="mw-headline" id="Properties">Properties</span></h2>
<p>For integer order α = <i>n</i>, <i>J</i><sub><i>n</i></sub> is often defined via a <a href="http://en.wikipedia.org/wiki/Laurent_series" title="Laurent series">Laurent series</a> for a generating function:</p>
<dl>
<dd><img class="tex" alt="e^{(x/2)(t-1/t)} = \sum_{n=-\infty}^\infty J_n(x) t^n,\!" src="Modified_Bessel_function_pliki/6e00e2420e10f2a9840eee2cb9d4a936.png"></dd>
</dl>
<p>an approach used by <a href="http://en.wikipedia.org/wiki/P._A._Hansen" title="P. A. Hansen" class="mw-redirect">P. A. Hansen</a> in 1843. (This can be generalized to non-integer order by <a href="http://en.wikipedia.org/wiki/Methods_of_contour_integration" title="Methods of contour integration">contour integration</a> or other methods.) Another important relation for integer orders is the <i><a href="http://en.wikipedia.org/wiki/Jacobi%E2%80%93Anger_expansion" title="Jacobi–Anger expansion">Jacobi–Anger expansion</a></i>:</p>
<dl>
<dd><img class="tex" alt="e^{iz \cos \phi} = \sum_{n=-\infty}^\infty i^n J_n(z) e^{in\phi},\!" src="Modified_Bessel_function_pliki/4142b7756ad80d68312f29b45633abe2.png"></dd>
</dl>
<p>and</p>
<dl>
<dd><img class="tex" alt="e^{iz \sin \phi} = \sum_{n=-\infty}^\infty J_n(z) e^{in\phi},\!" src="Modified_Bessel_function_pliki/452781926e4d1dfd28bbc9b6397a77fd.png"></dd>
</dl>
<p>which is used to expand a <a href="http://en.wikipedia.org/wiki/Plane_wave" title="Plane wave">plane wave</a> as a sum of cylindrical waves, or to find the <a href="http://en.wikipedia.org/wiki/Fourier_series" title="Fourier series">Fourier series</a> of a tone-modulated <a href="http://en.wikipedia.org/wiki/Frequency_modulation" title="Frequency modulation">FM</a> signal.</p>
<p>More generally, a series</p>
<dl>
<dd><img class="tex" alt="f(z)=a_0^\nu J_\nu (z)+ 2 \cdot \sum_{k=1} a_k^\nu J_{\nu+k}(z)\!" src="Modified_Bessel_function_pliki/78a85837466e0001576cc9602352faba.png"></dd>
</dl>
<p>is called Neumann expansion of <i>ƒ</i>. The coefficients for <img class="tex" alt="\nu=0" src="Modified_Bessel_function_pliki/1b7731e66bec242aca6183e5e1b517a7.png"> have the explicit form</p>
<dl>
<dd><img class="tex" alt="a_k^0=\frac{1}{2 \pi i} \int_{|z|=c} f(z) O_k(z) \, \mathrm d z,\!" src="Modified_Bessel_function_pliki/d2503d45b9862d859108796f021b7ead.png"></dd>
</dl>
<p>where <img class="tex" alt="O_k" src="Modified_Bessel_function_pliki/7bd86f8fb725973227d0fdebb7b4d59f.png"> is <a href="http://en.wikipedia.org/wiki/Neumann_polynomial" title="Neumann polynomial">Neumann's polynomial</a>.<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span>[</span>28<span>]</span></a></sup></p>
<p>Selected functions admit the special representation</p>
<dl>
<dd><img class="tex" alt="f(z)=\sum_{k=0} a_k^\nu J_{\nu+2k}(z)\!" src="Modified_Bessel_function_pliki/66b004b98c65603cad60df20fd550978.png"></dd>
</dl>
<p>with</p>
<dl>
<dd><img class="tex" alt="a_k^\nu=2(\nu+2k) \int_0^\infty f(z) \frac{J_{\nu+2k}(z)}z  \mathrm d z\!" src="Modified_Bessel_function_pliki/b987c79e69c64a40f0f0a4fa9cd83b42.png"></dd>
</dl>
<p>due to the orthogonality relation <img class="tex" alt="\int_0^\infty J_\alpha(z) J_\beta(z) \frac {\mathrm d z} z= \frac 2 \pi \frac{\sin\left(\frac \pi 2 (\alpha-\beta)  \right)}{\alpha^2 -\beta^2}." src="Modified_Bessel_function_pliki/93e402533c00d2a3b188adf1f8683fe8.png"></p>
<p>More generally, if <i>ƒ</i> has a branch-point near the origin of such a nature that <img class="tex" alt="f(z)= \sum_{k=0} a_k J_{\nu+k}(z)," src="Modified_Bessel_function_pliki/e009ca378b4890bf07ea2407efd06885.png"> then</p>
<dl>
<dd><img class="tex" alt="\mathcal{L} \left\{\sum_{k=0} a_k J_{\nu+k} \right\}(s)= \frac{1}{\sqrt{1+s^2}} \sum_{k=0} \frac{a_k}{(s+\sqrt{1+s^2})^{\nu+k}}" src="Modified_Bessel_function_pliki/16ba63f49c53d803762d32ae67c06c04.png"></dd>
</dl>
<p>or</p>
<dl>
<dd><img class="tex" alt="\sum_{k=0} a_k \xi^{\nu+k}= \frac{1+\xi^2}{2\xi} \mathcal L \{f \} \left( \frac{1-\xi^2}{2\xi} \right)," src="Modified_Bessel_function_pliki/575ffb514e3ddc13a3b67442e8e8fe98.png"></dd>
</dl>
<p>where <img class="tex" alt="\mathcal L \{f \}" src="Modified_Bessel_function_pliki/1c3bea37eee1a80233ab5f92154520e5.png"> is <i>ƒ</i>'s <a href="http://en.wikipedia.org/wiki/Laplace_transform" title="Laplace transform">Laplace transform</a>.<sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span>[</span>29<span>]</span></a></sup></p>
<p>Another way to define the Bessel functions is the Poisson representation formula and the Mehler-Sonine formula:</p>
<dl>
<dd><img class="tex" alt="\begin{align}J_\nu(z) &amp;= \frac{ (\frac{z}{2})^\nu }{ \Gamma(\nu + \frac{1}{2} ) \sqrt{\pi} } \int_{-1}^{1} e^{izs}(1 - s^2)^{\nu - \frac{1}{2} } ds, \\
&amp;=\frac 2{{\left(\frac z 2\right)}^\nu\cdot \sqrt{\pi} \cdot \Gamma\left(\frac 1 2-\nu\right)} \int_1^\infty  \frac{\sin(z u)}{(u^2-1)^{\nu+\frac 1 2}} d u,\end{align}" src="Modified_Bessel_function_pliki/0cb9a46efc2299a9db66d6f23afa2880.png"></dd>
</dl>
<p>where <i>ν</i>&nbsp;&gt;&nbsp;−1/2 and <i>z</i> is a complex number.<sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span>[</span>30<span>]</span></a></sup> This formula is useful especially when working with <a href="http://en.wikipedia.org/wiki/Fourier_transforms" title="Fourier transforms" class="mw-redirect">Fourier transforms</a>.</p>
<p>The functions <i>J</i><sub>α</sub>, <i>Y</i><sub>α</sub>, <i>H</i><sub>α</sub><sup>(1)</sup>, and <i>H</i><sub>α</sub><sup>(2)</sup> all satisfy the <a href="http://en.wikipedia.org/wiki/Recurrence_relation" title="Recurrence relation">recurrence relations</a>:</p>
<dl>
<dd><img class="tex" alt="\frac{2\alpha}{x} Z_\alpha(x) = Z_{\alpha-1}(x) + Z_{\alpha+1}(x)\!" src="Modified_Bessel_function_pliki/4034c70202d56a1f0914eb0d9eb8b5e4.png"></dd>
</dl>
<dl>
<dd><img class="tex" alt=" 2\frac{dZ_\alpha}{dx} = Z_{\alpha-1}(x) - Z_{\alpha+1}(x)\!" src="Modified_Bessel_function_pliki/499ab096e70c3772072d2182c14dd5ff.png"></dd>
</dl>
<p>where <i>Z</i> denotes <i>J</i>, <i>Y</i>, <i>H</i><sup>(1)</sup>, or <i>H</i><sup>(2)</sup>.
 (These two identities are often combined, e.g. added or subtracted, to 
yield various other relations.) In this way, for example, one can 
compute Bessel functions of higher orders (or higher derivatives) given 
the values at lower orders (or lower derivatives). In particular, it 
follows that:</p>
<dl>
<dd><img class="tex" alt="\left( \frac{d}{x dx} \right)^m \left[ x^\alpha Z_{\alpha} (x) \right] = x^{\alpha - m} Z_{\alpha - m} (x)" src="Modified_Bessel_function_pliki/d1a58356b3bf0462c7c606ccac45de73.png"></dd>
</dl>
<dl>
<dd><img class="tex" alt="\left( \frac{d}{x dx} \right)^m \left[ \frac{Z_\alpha (x)}{x^\alpha} \right] = (-1)^m \frac{Z_{\alpha + m} (x)}{x^{\alpha + m}}." src="Modified_Bessel_function_pliki/f4090bd15fe15edf10d603afa749b2e0.png"></dd>
</dl>
<p><i>Modified</i> Bessel functions follow similar relations&nbsp;:</p>
<dl>
<dd><img class="tex" alt="e^{(x/2)(t+1/t)} = \sum_{n=-\infty}^\infty I_n(x) t^n,\!" src="Modified_Bessel_function_pliki/85519cd3d7e7f381f5701d8b888417f8.png"></dd>
</dl>
<p>and</p>
<dl>
<dd><img class="tex" alt="e^{z \cos \theta} = I_0(z) + 2\sum_{n=1}^\infty  I_n(z) \cos(n\theta),\!" src="Modified_Bessel_function_pliki/35c78d96bca2d25f8130c278bbfb5a57.png"></dd>
</dl>
<p>The recurrence relation reads</p>
<dl>
<dd><img class="tex" alt="C_{\alpha-1}(x) - C_{\alpha+1}(x) = \frac{2\alpha}{x} C_\alpha(x)\!" src="Modified_Bessel_function_pliki/bbe6c79afa7eb82cbfa151fb9a163f83.png"></dd>
</dl>
<dl>
<dd><img class="tex" alt="C_{\alpha-1}(x) + C_{\alpha+1}(x) = 2\frac{dC_\alpha}{dx}\!" src="Modified_Bessel_function_pliki/2313c46c18b90a61644959e18c13547c.png"></dd>
</dl>
<p>where <i>C</i><sub>α</sub> denotes <i>I</i><sub>α</sub> or <i>e</i><sup>απ<i>i</i></sup><i>K</i><sub>α</sub>. These recurrence relations are useful for discrete diffusion problems.</p>
<p>Because Bessel's equation becomes <a href="http://en.wikipedia.org/wiki/Hermitian" title="Hermitian">Hermitian</a> (self-adjoint) if it is divided by <i>x</i>, the solutions must satisfy an orthogonality relationship for appropriate boundary conditions. In particular, it follows that:</p>
<dl>
<dd><img class="tex" alt="\int_0^1 x J_\alpha(x u_{\alpha,m}) J_\alpha(x u_{\alpha,n}) dx
= \frac{\delta_{m,n}}{2} [J_{\alpha+1}(u_{\alpha,m})]^2
= \frac{\delta_{m,n}}{2} [J_{\alpha}'(u_{\alpha,m})]^2,\!" src="Modified_Bessel_function_pliki/cdb1e8ba98f7855eba9777024cce03fd.png"></dd>
</dl>
<p>where <i>α</i>&nbsp;&gt;&nbsp;−1, δ<sub><i>m</i>,<i>n</i></sub> is the <a href="http://en.wikipedia.org/wiki/Kronecker_delta" title="Kronecker delta">Kronecker delta</a>, and <i>u</i><sub>α,m</sub> is the <i>m</i>-th <a href="http://en.wikipedia.org/wiki/Root_of_a_function" title="Root of a function" class="mw-redirect">zero</a> of <i>J</i><sub>α</sub>(<i>x</i>). This orthogonality relation can then be used to extract the coefficients in the <a href="http://en.wikipedia.org/wiki/Fourier%E2%80%93Bessel_series" title="Fourier–Bessel series">Fourier–Bessel series</a>, where a function is expanded in the basis of the functions <i>J</i><sub>α</sub>(<i>x</i> <i>u</i><sub>α,m</sub>) for fixed α and varying <i>m</i>.</p>
<p>An analogous relationship for the spherical Bessel functions follows immediately:</p>
<dl>
<dd><img class="tex" alt="\int_0^1 x^2 j_\alpha(x u_{\alpha,m}) j_\alpha(x u_{\alpha,n}) dx
= \frac{\delta_{m,n}}{2} [j_{\alpha+1}(u_{\alpha,m})]^2\!" src="Modified_Bessel_function_pliki/98df8d45fa6c659c53994ecde0481bbc.png"></dd>
</dl>
<p>Another orthogonality relation is the <i>closure equation</i>:<sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span>[</span>31<span>]</span></a></sup></p>
<dl>
<dd><img class="tex" alt="\int_0^\infty x J_\alpha(ux) J_\alpha(vx) dx = \frac{1}{u} \delta(u - v)\!" src="Modified_Bessel_function_pliki/0b0753caf104604740cc157746c856f5.png"></dd>
</dl>
<p>for <i>α</i>&nbsp;&gt;&nbsp;−1/2 and where δ is the <a href="http://en.wikipedia.org/wiki/Dirac_delta_function" title="Dirac delta function">Dirac delta function</a>. This property is used to construct an arbitrary function from a series of Bessel functions by means of the <a href="http://en.wikipedia.org/wiki/Hankel_transform" title="Hankel transform">Hankel transform</a>. For the spherical Bessel functions the orthogonality relation is:</p>
<dl>
<dd><img class="tex" alt="\int_0^\infty x^2 j_\alpha(ux) j_\alpha(vx) dx = \frac{\pi}{2u^2} \delta(u - v)\!" src="Modified_Bessel_function_pliki/b0db301791808108443b20fb78492c01.png"></dd>
</dl>
<p>for <i>α</i>&nbsp;&gt; &nbsp;−1.</p>
<p>Another important property of Bessel's equations, which follows from <a href="http://en.wikipedia.org/wiki/Abel%27s_identity" title="Abel's identity">Abel's identity</a>, involves the <a href="http://en.wikipedia.org/wiki/Wronskian" title="Wronskian">Wronskian</a> of the solutions:</p>
<dl>
<dd><img class="tex" alt="A_\alpha(x) \frac{dB_\alpha}{dx} - \frac{dA_\alpha}{dx} B_\alpha(x) = \frac{C_\alpha}{x},\!" src="Modified_Bessel_function_pliki/f0446689afdf22b84ad35c76d0f33caf.png"></dd>
</dl>
<p>where <i>A</i><sub>α</sub> and <i>B</i><sub>α</sub> are any two solutions of Bessel's equation, and <i>C</i><sub>α</sub> is a constant independent of <i>x</i> (which depends on α and on the particular Bessel functions considered). For example, if <i>A</i><sub>α</sub> = <i>J</i><sub>α</sub> and <i>B</i><sub>α</sub> = <i>Y</i><sub>α</sub>, then <i>C</i><sub>α</sub> is 2/π. This also holds for the modified Bessel functions; for example, if <i>A</i><sub>α</sub> = <i>I</i><sub>α</sub> and <i>B</i><sub>α</sub> = <i>K</i><sub>α</sub>, then <i>C</i><sub>α</sub> is&nbsp;−1.</p>
<p>(There are a large number of other known integrals and identities 
that are not reproduced here, but which can be found in the references.)</p>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Bessel_function&amp;action=edit&amp;section=17" title="Edit section: Multiplication theorem">edit</a>]</span> <span class="mw-headline" id="Multiplication_theorem">Multiplication theorem</span></h2>
<p>The Bessel functions obey a <a href="http://en.wikipedia.org/wiki/Multiplication_theorem" title="Multiplication theorem">multiplication theorem</a></p>
<dl>
<dd><img class="tex" alt="\lambda^{-\nu} J_\nu (\lambda z) =
\sum_{n=0}^\infty \frac{1}{n!}
\left(\frac{(1-\lambda^2)z}{2}\right)^n
J_{\nu+n}(z)
" src="Modified_Bessel_function_pliki/2667d0803affc1a7f2ffef52949c25a9.png"></dd>
</dl>
<p>where <img class="tex" alt="\lambda" src="Modified_Bessel_function_pliki/e05a30d96800384dd38b22851322a6b5.png"> and <img class="tex" alt="\nu" src="Modified_Bessel_function_pliki/7368318dd3647eb6bbf6afaf6d26c48d.png"> may be taken as arbitrary complex numbers. A similar form may be given for <img class="tex" alt="Y_\nu(z)" src="Modified_Bessel_function_pliki/5f1b502f705ac4d6ec8b8c2f00300813.png"> and so on<sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span>[</span>32<span>]</span></a></sup> <sup id="cite_ref-32" class="reference"><a href="#cite_note-32"><span>[</span>33<span>]</span></a></sup></p>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Bessel_function&amp;action=edit&amp;section=18" title="Edit section: Bourget's hypothesis">edit</a>]</span> <span class="mw-headline" id="Bourget.27s_hypothesis">Bourget's hypothesis</span></h2>
<p>Bessel himself originally proved that for non-negative integers <i>n</i>, the equation <i>J</i><sub><i>n</i></sub>(<i>x</i>)&nbsp;=&nbsp;0 has an infinite number of solutions in <i>x</i>.<sup id="cite_ref-33" class="reference"><a href="#cite_note-33"><span>[</span>34<span>]</span></a></sup> When the functions <i>J</i><sub><i>n</i></sub>(<i>x</i>) are plotted on the same graph, though, none of the zeros seem to coincide for different values of <i>n</i> except for the zero at <i>x</i>&nbsp;=&nbsp;0. This phenomenon is known as <b>Bourget's hypothesis</b> after the nineteenth century French mathematician who studied Bessel functions. Specifically it states that for any integers <i>n</i>&nbsp;≥&nbsp;0 and <i>m</i>&nbsp;≥&nbsp;1, the functions <i>J</i><sub><i>n</i></sub>(<i>x</i>) and <i>J</i><sub><i>n</i>+<i>m</i></sub>(<i>x</i>) have no common zeros other than the one at <i>x</i>&nbsp;=&nbsp;0. The theorem was proved by Siegel in 1929.<sup id="cite_ref-34" class="reference"><a href="#cite_note-34"><span>[</span>35<span>]</span></a></sup></p>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Bessel_function&amp;action=edit&amp;section=19" title="Edit section: Selected identities">edit</a>]</span> <span class="mw-headline" id="Selected_identities">Selected identities</span></h2>
<p><sup id="cite_ref-35" class="reference"><a href="#cite_note-35"><span>[</span>36<span>]</span></a></sup></p>
<ul>
<li><img class="tex" alt="I_{-1/2} \left(z\right)= \sqrt{\frac{2}{\pi z}}\cosh(z)&nbsp;;" src="Modified_Bessel_function_pliki/7605c4a98cbad00eaceae22c210ad389.png"></li>
<li><img class="tex" alt="I_{1/2} \left(z\right)= \sqrt{\frac{2}{\pi z}}\sinh(z)&nbsp;;" src="Modified_Bessel_function_pliki/a093ac6c077be5f986dab9e016a67e65.png"></li>
<li><img class="tex" alt="I_\nu(z)=\sum_{k=0} \frac{z^k}{k!} J_{\nu+k}(z);" src="Modified_Bessel_function_pliki/e82c08746359ab8eeb55cf73bdd84d74.png"></li>
<li><img class="tex" alt="J_\nu(z)=\sum_{k=0} (-1)^k \frac{z^k}{k!} I_{\nu+k}(z);" src="Modified_Bessel_function_pliki/d489937d3afaedf508a5eaa3ad465dc5.png"></li>
<li><img class="tex" alt="I_\nu (\lambda z)= \lambda^\nu \sum_{k=0} \frac{\left((\lambda^2-1)\frac z 2\right)^k}{k!} I_{\nu+k}(z);" src="Modified_Bessel_function_pliki/d7f023a0d71d6b1039afb4f0c06e5c7f.png"></li>
<li><img class="tex" alt="I_\nu (z_1+z_2)= \sum_{k=-\infty}^\infty I_{\nu-k}(z_1)I_k(z_2),\quad J_\nu(z_1\pm z_2)= \sum_{k=-\infty}^\infty J_{\nu \mp k}(z_1)J_k(z_2);" src="Modified_Bessel_function_pliki/bbbcc8430a903658035ead3b4caa2413.png"></li>
<li><img class="tex" alt="J_\nu(z)=\frac z {2 \nu} (J_{\nu-1}(z)+J_{\nu+1}(z)), \quad I_\nu(z)=\frac z {2 \nu} (I_{\nu-1}(z)-I_{\nu+1}(z));" src="Modified_Bessel_function_pliki/5463696847421c0f22d9fe189bfc6474.png"></li>
<li><img class="tex" alt="J_\nu'(z)=\tfrac{1}{2} (J_{\nu-1}(z)-J_{\nu+1}(z)), \quad I_\nu'(z)=\tfrac{1}{2}(I_{\nu-1}(z)+I_{\nu+1}(z));" src="Modified_Bessel_function_pliki/7dbe712e98444e32bf90cad70c8e76cb.png"></li>
<li><img class="tex" alt="\left(\tfrac{1}{2}z\right)^\nu= \Gamma(\nu)\cdot \sum_{k=0} I_{\nu+2k}(z)(\nu+2k){-\nu\choose k}
= \Gamma(\nu)\cdot\sum_{k=0}(-1)^k J_{\nu+2k}(z)(\nu+2k){-\nu \choose k}
= \Gamma(\nu+1)\cdot \sum_{k=0}\frac 1{k!}\left(\tfrac1 2z\right)^k J_{\nu+k}(z)." src="Modified_Bessel_function_pliki/cbffcd91578cb66cd0818dab3e4b2ce6.png"></li>
<li><img class="tex" alt="K_\frac{1}{2}(z)=\sqrt{\frac{\pi}{2}} \mathrm{e}^{-z}z^{-1/2},\, z&gt;0 " src="Modified_Bessel_function_pliki/5dae74567e273d78f414d8005d818995.png"></li>
</ul>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Bessel_function&amp;action=edit&amp;section=20" title="Edit section: See also">edit</a>]</span> <span class="mw-headline" id="See_also">See also</span></h2>
<ul>
<li><a href="http://en.wikipedia.org/wiki/Bessel%E2%80%93Clifford_function" title="Bessel–Clifford function">Bessel–Clifford function</a></li>
<li><a href="http://en.wikipedia.org/wiki/Bessel_polynomials" title="Bessel polynomials">Bessel polynomials</a></li>
<li><a href="http://en.wikipedia.org/wiki/Propagator" title="Propagator">Propagator</a></li>
<li><a href="http://en.wikipedia.org/wiki/Fourier%E2%80%93Bessel_series" title="Fourier–Bessel series">Fourier–Bessel series</a></li>
<li><a href="http://en.wikipedia.org/wiki/Hahn%E2%80%93Exton_q-Bessel_function" title="Hahn–Exton q-Bessel function">Hahn–Exton q-Bessel function</a></li>
<li><a href="http://en.wikipedia.org/wiki/Jackson_q-Bessel_function" title="Jackson q-Bessel function">Jackson q-Bessel function</a></li>
<li><a href="http://en.wikipedia.org/wiki/Struve_function" title="Struve function">Struve function</a></li>
<li><a href="http://en.wikipedia.org/wiki/Kelvin_functions" title="Kelvin functions">Kelvin functions</a></li>
<li><a href="http://en.wikipedia.org/wiki/Lommel_function" title="Lommel function">Lommel functions</a></li>
<li><a href="http://en.wikipedia.org/wiki/Lommel_polynomial" title="Lommel polynomial">Lommel polynomial</a></li>
<li><a href="http://en.wikipedia.org/wiki/Neumann_polynomial" title="Neumann polynomial">Neumann polynomial</a></li>
<li><a href="http://en.wikipedia.org/wiki/Vibrations_of_a_circular_drum" title="Vibrations of a circular drum" class="mw-redirect">Vibrations of a circular drum</a></li>
<li><a href="http://en.wikipedia.org/wiki/Wright_generalized_Bessel_function" title="Wright generalized Bessel function" class="mw-redirect">Wright generalized Bessel function</a></li>
<li><a href="http://en.wikipedia.org/wiki/Frequency_Modulation" title="Frequency Modulation" class="mw-redirect">Frequency Modulation</a></li>
</ul>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Bessel_function&amp;action=edit&amp;section=21" title="Edit section: Notes">edit</a>]</span> <span class="mw-headline" id="Notes">Notes</span></h2>
<div class="reflist references-column-width" style="-moz-column-width: 30em; -webkit-column-width: 30em; column-width: 30em; list-style-type: decimal;">
<ol class="references">
<li id="cite_note-0"><b><a href="#cite_ref-0">^</a></b> <span class="reference-text">Abramowitz and Stegun, <a rel="nofollow" class="external text" href="http://www.math.sfu.ca/%7Ecbm/aands/page_360.htm">p. 360, 9.1.10</a>.</span></li>
<li id="cite_note-1"><b><a href="#cite_ref-1">^</a></b> <span class="reference-text">Abramowitz and Stegun, <a rel="nofollow" class="external text" href="http://www.math.sfu.ca/%7Ecbm/aands/page_358.htm">p. 358, 9.1.5</a>.</span></li>
<li id="cite_note-2"><b><a href="#cite_ref-2">^</a></b> <span class="reference-text">Watson, <a rel="nofollow" class="external text" href="http://books.google.com/books?id=Mlk3FrNoEVoC&amp;pg=PA176">p. 176</a></span></li>
<li id="cite_note-3"><b><a href="#cite_ref-3">^</a></b> <span class="reference-text"><a rel="nofollow" class="external free" href="http://www.math.ohio-state.edu/%7Egerlach/math/BVtypset/node122.html">http://www.math.ohio-state.edu/~gerlach/math/BVtypset/node122.html</a></span></li>
<li id="cite_note-4"><b><a href="#cite_ref-4">^</a></b> <span class="reference-text"><a rel="nofollow" class="external free" href="http://www.nbi.dk/%7Epolesen/borel/node15.html">http://www.nbi.dk/~polesen/borel/node15.html</a></span></li>
<li id="cite_note-5"><b><a href="#cite_ref-5">^</a></b> <span class="reference-text">Arfken &amp; Weber, exercise 11.1.17.</span></li>
<li id="cite_note-6"><b><a href="#cite_ref-6">^</a></b> <span class="reference-text">Szegö, G. Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., 1975.</span></li>
<li id="cite_note-7"><b><a href="#cite_ref-7">^</a></b> <span class="reference-text">Watson, <a rel="nofollow" class="external text" href="http://books.google.com/books?id=Mlk3FrNoEVoC&amp;pg=PA178">p. 178</a>.</span></li>
<li id="cite_note-8"><b><a href="#cite_ref-8">^</a></b> <span class="reference-text">Abramowitz and Stegun, <a rel="nofollow" class="external text" href="http://www.math.sfu.ca/%7Ecbm/aands/page_358.htm">p. 358, 9.1.3, 9.1.4</a>.</span></li>
<li id="cite_note-9"><b><a href="#cite_ref-9">^</a></b> <span class="reference-text">Abramowitz and Stegun, <a rel="nofollow" class="external text" href="http://www.math.sfu.ca/%7Ecbm/aands/page_358.htm">p. 358, 9.1.6</a>.</span></li>
<li id="cite_note-10"><b><a href="#cite_ref-10">^</a></b> <span class="reference-text">Abramowitz and Stegun, <a rel="nofollow" class="external text" href="http://www.math.sfu.ca/%7Ecbm/aands/page_360.htm">p. 360, 9.1.25</a>.</span></li>
<li id="cite_note-11"><b><a href="#cite_ref-11">^</a></b> <span class="reference-text">Watson, <a rel="nofollow" class="external text" href="http://books.google.com/books?id=Mlk3FrNoEVoC&amp;pg=PA178">p. 178</a></span></li>
<li id="cite_note-12"><b><a href="#cite_ref-12">^</a></b> <span class="reference-text">Abramowitz and Stegun, <a rel="nofollow" class="external text" href="http://www.math.sfu.ca/%7Ecbm/aands/page_375.htm">p. 375, 9.6.2, 9.6.10, 9.6.11</a>.</span></li>
<li id="cite_note-13"><b><a href="#cite_ref-13">^</a></b> <span class="reference-text">Abramowitz and Stegun, <a rel="nofollow" class="external text" href="http://www.math.sfu.ca/%7Ecbm/aands/page_374.htm">p. 374, 9.6.1</a>.</span></li>
<li id="cite_note-14"><b><a href="#cite_ref-14">^</a></b> <span class="reference-text">Watson, <a rel="nofollow" class="external text" href="http://books.google.com/books?id=Mlk3FrNoEVoC&amp;pg=PA181">p. 181</a>.</span></li>
<li id="cite_note-15"><b><a href="#cite_ref-15">^</a></b> <span class="reference-text">M.Kh.Khokonov. <i>Cascade Processes of Energy Loss by Emission of Hard Photons</i>, JETP, V.99, No.4, pp. 690-707 (2004). Derived from formulas sourced to I. S. Gradshteĭn and I. M. Ryzhik, <i>Table of Integrals, Series, and Products</i> (Fizmatgiz, Moscow, 1963; Academic, New York, 1980).</span></li>
<li id="cite_note-16"><b><a href="#cite_ref-16">^</a></b> <span class="reference-text">Referred to as such in: Teichroew, D. <i>The Mixture of Normal Distributions with Different Variances</i>, The Annals of Mathematical Statistics. Vol. 28, No. 2 (Jun., 1957), pp. 510–512</span></li>
<li id="cite_note-17"><b><a href="#cite_ref-17">^</a></b> <span class="reference-text">Abramowitz and Stegun, <a rel="nofollow" class="external text" href="http://www.math.sfu.ca/%7Ecbm/aands/page_437.htm">p. 437, 10.1.1</a>.</span></li>
<li id="cite_note-18"><b><a href="#cite_ref-18">^</a></b> <span class="reference-text">Abramowitz and Stegun, <a rel="nofollow" class="external text" href="http://www.math.sfu.ca/%7Ecbm/aands/page_439.htm">p. 439, 10.1.25, 10.1.26</a>;</span></li>
<li id="cite_note-19"><b><a href="#cite_ref-19">^</a></b> <span class="reference-text">Abramowitz and Stegun, <a rel="nofollow" class="external text" href="http://www.math.sfu.ca/%7Ecbm/aands/page_438.htm">p. 438, 10.1.11</a>.</span></li>
<li id="cite_note-20"><b><a href="#cite_ref-20">^</a></b> <span class="reference-text">Abramowitz and Stegun, <a rel="nofollow" class="external text" href="http://www.math.sfu.ca/%7Ecbm/aands/page_438.htm">p. 438, 10.1.12</a>;</span></li>
<li id="cite_note-21"><b><a href="#cite_ref-21">^</a></b> <span class="reference-text">Abramowitz and Stegun, <a rel="nofollow" class="external text" href="http://www.math.sfu.ca/%7Ecbm/aands/page_439.htm">p. 439, 10.1.39</a>.</span></li>
<li id="cite_note-22"><b><a href="#cite_ref-22">^</a></b> <span class="reference-text">Abramowitz and Stegun, <a rel="nofollow" class="external text" href="http://www.math.sfu.ca/%7Ecbm/aands/page_439.htm">p. 439, 10.1.23</a>.</span></li>
<li id="cite_note-23"><b><a href="#cite_ref-23">^</a></b> <span class="reference-text">Hong Du, "Mie-scattering calculation," <i>Applied Optics</i> <b>43</b> (9), 1951–1956 (2004)</span></li>
<li id="cite_note-Arfken_.26_Weber-24">^ <a href="#cite_ref-Arfken_.26_Weber_24-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Arfken_.26_Weber_24-1"><sup><i><b>b</b></i></sup></a> <span class="reference-text">Arfken &amp; Weber.</span></li>
<li id="cite_note-25"><b><a href="#cite_ref-25">^</a></b> <span class="reference-text">Abramowitz and Stegun, <a rel="nofollow" class="external text" href="http://www.math.sfu.ca/%7Ecbm/aands/page_377.htm">p. 377, 9.7.1</a>;</span></li>
<li id="cite_note-26"><b><a href="#cite_ref-26">^</a></b> <span class="reference-text">Abramowitz and Stegun, <a rel="nofollow" class="external text" href="http://www.math.sfu.ca/%7Ecbm/aands/page_378.htm">p. 378, 9.7.2</a>;</span></li>
<li id="cite_note-27"><b><a href="#cite_ref-27">^</a></b> <span class="reference-text">Abramowitz and Stegun, <a rel="nofollow" class="external text" href="http://www.math.sfu.ca/%7Ecbm/aands/page_363.htm">p. 363, 9.1.82</a> ff.</span></li>
<li id="cite_note-28"><b><a href="#cite_ref-28">^</a></b> <span class="reference-text"><a rel="nofollow" class="external text" href="http://books.google.com/books?id=Mlk3FrNoEVoC&amp;lpg=PA522&amp;ots=SOShEJmay6&amp;dq=bessel%20neumann%20series&amp;hl=de&amp;pg=PA536#v=onepage&amp;q=bessel%20neumann%20series&amp;f=false">E. T. Whittaker, G. N. Watson, A course in modern Analysis p. 536</a></span></li>
<li id="cite_note-29"><b><a href="#cite_ref-29">^</a></b> <span class="reference-text">I.S. Gradshteyn (И.С. Градштейн), I.M. Ryzhik (И.М. Рыжик); Alan Jeffrey, Daniel Zwillinger, editors. <i>Table of Integrals, Series, and Products</i>, seventh edition. Academic Press, 2007. <a href="http://en.wikipedia.org/wiki/Special:BookSources/9780123736376" class="internal mw-magiclink-isbn">ISBN 978-0-12-373637-6</a>. Equation 8.411.10</span></li>
<li id="cite_note-30"><b><a href="#cite_ref-30">^</a></b> <span class="reference-text">Arfken &amp; Weber, section 11.2</span></li>
<li id="cite_note-31"><b><a href="#cite_ref-31">^</a></b> <span class="reference-text">Abramowitz and Stegun, <a rel="nofollow" class="external text" href="http://www.math.sfu.ca/%7Ecbm/aands/page_363.htm">p. 363, 9.1.74</a>.</span></li>
<li id="cite_note-32"><b><a href="#cite_ref-32">^</a></b> <span class="reference-text">C. Truesdell, "<a rel="nofollow" class="external text" href="http://www.pnas.org/cgi/reprint/36/12/752.pdf">On the Addition and Multiplication Theorems for the Special Functions</a>", <i>Proceedings of the National Academy of Sciences, Mathematics</i>, (1950) pp.752–757.</span></li>
<li id="cite_note-33"><b><a href="#cite_ref-33">^</a></b> <span class="reference-text">F. Bessel, <i>Untersuchung des Theils der planetarischen Störungen</i>, Berlin Abhandlungen (1824), article 14.</span></li>
<li id="cite_note-34"><b><a href="#cite_ref-34">^</a></b> <span class="reference-text">Watson, pp. 484–5</span></li>
<li id="cite_note-35"><b><a href="#cite_ref-35">^</a></b> <span class="reference-text">See, for example, Lide DR. CRC handbook of chemistry and physics: a ready-reference book of chemical CRC Press, 2004, <a href="http://en.wikipedia.org/wiki/Special:BookSources/0849304857" class="internal mw-magiclink-isbn">ISBN 0849304857</a>, p. A-95</span></li>
</ol>
</div>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Bessel_function&amp;action=edit&amp;section=22" title="Edit section: References">edit</a>]</span> <span class="mw-headline" id="References">References</span></h2>
<ul>
<li><span class="citation" id="CITEREFAbramowitzStegun1965">Abramowitz, Milton; Stegun, Irene A., eds. (1965), <a rel="nofollow" class="external text" href="http://www.math.sfu.ca/%7Ecbm/aands/page_355.htm">"Chapter 9"</a>, <i><a href="http://en.wikipedia.org/wiki/Abramowitz_and_Stegun" title="Abramowitz and Stegun">Handbook of Mathematical Functions</a> with Formulas, Graphs, and Mathematical Tables</i>, New York: Dover, pp.&nbsp;355, <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&nbsp;<a href="http://en.wikipedia.org/wiki/Special:BookSources/978-0486612720" title="Special:BookSources/978-0486612720">978-0486612720</a>, <a href="http://en.wikipedia.org/wiki/Mathematical_Reviews" title="Mathematical Reviews">MR</a><a rel="nofollow" class="external text" href="http://www.ams.org/mathscinet-getitem?mr=0167642">0167642</a><span class="printonly">, <a rel="nofollow" class="external free" href="http://www.math.sfu.ca/%7Ecbm/aands/page_355.htm">http://www.math.sfu.ca/~cbm/aands/page_355.htm</a></span></span><span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=bookitem&amp;rft.btitle=Chapter+9&amp;rft.atitle=%5B%5BAbramowitz+and+Stegun%7CHandbook+of+Mathematical+Functions%5D%5D+with+Formulas%2C+Graphs%2C+and+Mathematical+Tables&amp;rft.date=1965&amp;rft.pages=pp.%26nbsp%3B355&amp;rft.place=New+York&amp;rft.pub=Dover&amp;rft.isbn=978-0486612720&amp;rft.mr=0167642&amp;rft_id=&amp;rfr_id=info:sid/en.wikipedia.org:Bessel_function"><span style="display: none;">&nbsp;</span></span> See also <a rel="nofollow" class="external text" href="http://www.math.sfu.ca/%7Ecbm/aands/page_435.htm">chapter 10</a>.</li>
<li>Arfken, George B. and Hans J. Weber, <i>Mathematical Methods for Physicists</i>, 6th edition (Harcourt: San Diego, 2005). <a href="http://en.wikipedia.org/wiki/Special:BookSources/0120598760" class="internal mw-magiclink-isbn">ISBN 0-12-059876-0</a>.</li>
<li>Bayin, S.S. <i>Mathematical Methods in Science and Engineering</i>, Wiley, 2006, Chapter 6.</li>
<li>Bayin, S.S., <i>Essentials of Mathematical Methods in Science and Engineering</i>, Wiley, 2008, Chapter 11.</li>
<li>Bowman, Frank <i>Introduction to Bessel Functions</i> (Dover: New York, 1958). <a href="http://en.wikipedia.org/wiki/Special:BookSources/0486604624" class="internal mw-magiclink-isbn">ISBN 0-486-60462-4</a>.</li>
<li>G. Mie, "Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen", <i>Ann. Phys. Leipzig</i> <b>25</b> (1908), p.&nbsp;377.</li>
<li><span class="citation" id="CITEREFOlverMaximon2010">Olver, F. W. J.; Maximon, L. C. (2010), <a rel="nofollow" class="external text" href="http://dlmf.nist.gov/10">"Bessel function"</a>, in <a href="http://en.wikipedia.org/wiki/Frank_W._J._Olver" title="Frank W. J. Olver">Olver, Frank W. J.</a>; Lozier, Daniel M.; Boisvert, Ronald F. et al., <i><a href="http://en.wikipedia.org/wiki/Digital_Library_of_Mathematical_Functions" title="Digital Library of Mathematical Functions">NIST Handbook of Mathematical Functions</a></i>, Cambridge University Press, <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&nbsp;<a href="http://en.wikipedia.org/wiki/Special:BookSources/978-0521192255" title="Special:BookSources/978-0521192255">978-0521192255</a>, <a href="http://en.wikipedia.org/wiki/Mathematical_Reviews" title="Mathematical Reviews">MR</a><a rel="nofollow" class="external text" href="http://www.ams.org/mathscinet-getitem?mr=2723248">2723248</a><span class="printonly">, <a rel="nofollow" class="external free" href="http://dlmf.nist.gov/10">http://dlmf.nist.gov/10</a></span></span><span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=bookitem&amp;rft.btitle=Bessel+function&amp;rft.atitle=%5B%5BDigital+Library+of+Mathematical+Functions%7CNIST+Handbook+of+Mathematical+Functions%5D%5D&amp;rft.aulast=Olver&amp;rft.aufirst=F.+W.+J.&amp;rft.au=Olver%2C%26%2332%3BF.+W.+J.&amp;rft.au=Maximon%2C%26%2332%3BL.+C.&amp;rft.date=2010&amp;rft.pub=Cambridge+University+Press&amp;rft.isbn=978-0521192255&amp;rft_id=&amp;rfr_id=info:sid/en.wikipedia.org:Bessel_function"><span style="display: none;">&nbsp;</span></span></li>
<li><span class="citation" id="CITEREFPressTeukolskyVetterlingFlannery2007">Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007), <a rel="nofollow" class="external text" href="http://apps.nrbook.com/empanel/index.html#pg=274">"Section 6.5. Bessel Functions of Integer Order"</a>, <i>Numerical Recipes: The Art of Scientific Computing</i> (3rd ed.), New York: Cambridge University Press, <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&nbsp;<a href="http://en.wikipedia.org/wiki/Special:BookSources/978-0-521-88068-8" title="Special:BookSources/978-0-521-88068-8">978-0-521-88068-8</a><span class="printonly">, <a rel="nofollow" class="external free" href="http://apps.nrbook.com/empanel/index.html#pg=274">http://apps.nrbook.com/empanel/index.html#pg=274</a></span></span><span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=bookitem&amp;rft.btitle=Section+6.5.+Bessel+Functions+of+Integer+Order&amp;rft.atitle=Numerical+Recipes%3A+The+Art+of+Scientific+Computing&amp;rft.aulast=Press&amp;rft.aufirst=WH&amp;rft.au=Press%2C%26%2332%3BWH&amp;rft.au=Teukolsky%2C%26%2332%3BSA&amp;rft.au=Vetterling%2C%26%2332%3BWT&amp;rft.au=Flannery%2C%26%2332%3BBP&amp;rft.date=2007&amp;rft.edition=3rd&amp;rft.place=New+York&amp;rft.pub=Cambridge+University+Press&amp;rft.isbn=978-0-521-88068-8&amp;rft_id=&amp;rfr_id=info:sid/en.wikipedia.org:Bessel_function"><span style="display: none;">&nbsp;</span></span></li>
<li>B Spain, M.G. Smith, <i>Functions of mathematical physics</i>, Van Nostrand Reinhold Company, London, 1970. Chapter 9 deals with Bessel functions.</li>
<li>N. M. Temme, <i>Special Functions. An Introduction to the Classical Functions of Mathematical Physics</i>, John Wiley and Sons, Inc., New York, 1996. <a href="http://en.wikipedia.org/wiki/Special:BookSources/0471113131" class="internal mw-magiclink-isbn">ISBN 0-471-11313-1</a>. Chapter 9 deals with Bessel functions.</li>
<li><a href="http://en.wikipedia.org/wiki/G._N._Watson" title="G. N. Watson">Watson, G.N.</a>, <i>A Treatise on the Theory of Bessel Functions, Second Edition</i>, (1995) Cambridge University Press. <a href="http://en.wikipedia.org/wiki/Special:BookSources/0521483913" class="internal mw-magiclink-isbn">ISBN 0-521-48391-3</a>.</li>
</ul>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Bessel_function&amp;action=edit&amp;section=23" title="Edit section: External links">edit</a>]</span> <span class="mw-headline" id="External_links">External links</span></h2>
<ul>
<li><span class="citation" id="CITEREFLizorkin2001">Lizorkin, P.I. (2001), <a rel="nofollow" class="external text" href="http://www.encyclopediaofmath.org/index.php?title=b/b015840">"Bessel functions"</a>, in Hazewinkel, Michiel, <i><a href="http://en.wikipedia.org/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, Springer, <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&nbsp;<a href="http://en.wikipedia.org/wiki/Special:BookSources/978-1556080104" title="Special:BookSources/978-1556080104">978-1556080104</a><span class="printonly">, <a rel="nofollow" class="external free" href="http://www.encyclopediaofmath.org/index.php?title=b/b015840">http://www.encyclopediaofmath.org/index.php?title=b/b015840</a></span></span><span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=bookitem&amp;rft.btitle=Bessel+functions&amp;rft.atitle=%5B%5BEncyclopedia+of+Mathematics%5D%5D&amp;rft.aulast=Lizorkin&amp;rft.aufirst=P.I.&amp;rft.au=Lizorkin%2C%26%2332%3BP.I.&amp;rft.date=2001&amp;rft.pub=Springer&amp;rft.isbn=978-1556080104&amp;rft_id=&amp;rfr_id=info:sid/en.wikipedia.org:Bessel_function"><span style="display: none;">&nbsp;</span></span></li>
<li><span class="citation" id="CITEREFKarmazinaPrudnikov2001">Karmazina, L.N.; Prudnikov, A.P. (2001), <a rel="nofollow" class="external text" href="http://www.encyclopediaofmath.org/index.php?title=c/c027610">"Cylinder function"</a>, in Hazewinkel, Michiel, <i><a href="http://en.wikipedia.org/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, Springer, <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&nbsp;<a href="http://en.wikipedia.org/wiki/Special:BookSources/978-1556080104" title="Special:BookSources/978-1556080104">978-1556080104</a><span class="printonly">, <a rel="nofollow" class="external free" href="http://www.encyclopediaofmath.org/index.php?title=c/c027610">http://www.encyclopediaofmath.org/index.php?title=c/c027610</a></span></span><span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=bookitem&amp;rft.btitle=Cylinder+function&amp;rft.atitle=%5B%5BEncyclopedia+of+Mathematics%5D%5D&amp;rft.aulast=Karmazina&amp;rft.aufirst=L.N.&amp;rft.au=Karmazina%2C%26%2332%3BL.N.&amp;rft.au=Prudnikov%2C%26%2332%3BA.P.&amp;rft.date=2001&amp;rft.pub=Springer&amp;rft.isbn=978-1556080104&amp;rft_id=&amp;rfr_id=info:sid/en.wikipedia.org:Bessel_function"><span style="display: none;">&nbsp;</span></span></li>
<li><span class="citation" id="CITEREFRozov2001">Rozov, =N.Kh. (2001), <a rel="nofollow" class="external text" href="http://www.encyclopediaofmath.org/index.php?title=B/b015830">"Bessel equation"</a>, in Hazewinkel, Michiel, <i><a href="http://en.wikipedia.org/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, Springer, <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&nbsp;<a href="http://en.wikipedia.org/wiki/Special:BookSources/978-1556080104" title="Special:BookSources/978-1556080104">978-1556080104</a><span class="printonly">, <a rel="nofollow" class="external free" href="http://www.encyclopediaofmath.org/index.php?title=B/b015830">http://www.encyclopediaofmath.org/index.php?title=B/b015830</a></span></span><span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=bookitem&amp;rft.btitle=Bessel+equation&amp;rft.atitle=%5B%5BEncyclopedia+of+Mathematics%5D%5D&amp;rft.aulast=Rozov&amp;rft.aufirst=%3DN.Kh.&amp;rft.au=Rozov%2C%26%2332%3B%3DN.Kh.&amp;rft.date=2001&amp;rft.pub=Springer&amp;rft.isbn=978-1556080104&amp;rft_id=&amp;rfr_id=info:sid/en.wikipedia.org:Bessel_function"><span style="display: none;">&nbsp;</span></span></li>
<li>Wolfram function pages on Bessel <a rel="nofollow" class="external text" href="http://functions.wolfram.com/Bessel-TypeFunctions/BesselJ/">J</a> and <a rel="nofollow" class="external text" href="http://functions.wolfram.com/Bessel-TypeFunctions/BesselY/">Y</a> functions, and modified Bessel <a rel="nofollow" class="external text" href="http://functions.wolfram.com/Bessel-TypeFunctions/BesselI/">I</a> and <a rel="nofollow" class="external text" href="http://functions.wolfram.com/Bessel-TypeFunctions/BesselK/">K</a> functions. Pages include formulas, function evaluators, and plotting calculators.</li>
<li><a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/BesselFunctionoftheFirstKind.html">Wolfram Mathworld – Bessel functions of the first kind</a></li>
<li>Bessel functions <a rel="nofollow" class="external text" href="http://www.librow.com/articles/article-11/appendix-a-34">J<sub>ν</sub></a>, <a rel="nofollow" class="external text" href="http://www.librow.com/articles/article-11/appendix-a-35">Y<sub>ν</sub></a>, <a rel="nofollow" class="external text" href="http://www.librow.com/articles/article-11/appendix-a-36">I<sub>ν</sub></a> and <a rel="nofollow" class="external text" href="http://www.librow.com/articles/article-11/appendix-a-37">K<sub>ν</sub></a> in Librow <a rel="nofollow" class="external text" href="http://www.librow.com/articles/article-11">Function handbook</a>.</li>
</ul>


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